diff --git "a/minictx-valid/htpi.jsonl" "b/minictx-valid/htpi.jsonl" new file mode 100644--- /dev/null +++ "b/minictx-valid/htpi.jsonl" @@ -0,0 +1,45 @@ +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\n", "theoremStatement": "theorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a ", "theoremName": "HTPI.Theorem_7_3_7", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", 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"Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Meta.Tactic.Delta", "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.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "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.Meta.Tactic.Symm", "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.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "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", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done", "proofType": "tactic", "proofLengthLines": 26, "proofLengthTokens": 807}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\n", "theoremStatement": "lemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) ", "theoremName": "HTPI.G_maps_below", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", 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"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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 204}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap5\nnamespace HTPI\n\n/- Definitions -/\ndef invRel {U V : Type} (R : Rel U V) : Rel V U :=\n RelFromExt (inv (extension R))\n\ndef rel_within {U V : Type} (R : Rel U V) (A : Set U) (B : Set V) : Prop :=\n \u2200 \u2983x : U\u2984 \u2983y : V\u2984, R x y \u2192 x \u2208 A \u2227 y \u2208 B\n\ndef fcnl_on {U V : Type} (R : Rel U V) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203! (y : V), R x y\n\ndef matching {U V : Type} (R : Rel U V) (A : Set U) (B : Set V) : Prop :=\n rel_within R A B \u2227 fcnl_on R A \u2227 fcnl_on (invRel R) B\n\ndef equinum {U V : Type} (A : Set U) (B : Set V) : Prop :=\n \u2203 (R : Rel U V), matching R A B\n\nnotation:50 A:50 \" \u223c \" B:50 => equinum A B\n\ndef RelWithinFromFunc {U V : Type} (f : U \u2192 V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 f x = y\n\ndef one_one_on {U V : Type} (f : U \u2192 V) (A : Set U) : Prop :=\n \u2200 \u2983x1 x2 : U\u2984, x1 \u2208 A \u2192 x2 \u2208 A \u2192 f x1 = f x2 \u2192 x1 = x2\n\ndef Univ (U : Type) : Set U := {x : U | True}\n\ndef compRel {U V W : Type} (S : Rel V W) (R : Rel U V) : Rel U W :=\n RelFromExt (comp (extension S) (extension R))\n\ndef I (n : Nat) : Set Nat := {k : Nat | k < n}\n\ndef finite {U : Type} (A : Set U) : Prop :=\n \u2203 (n : Nat), I n \u223c A\n\ndef denum {U : Type} (A : Set U) : Prop :=\n Univ Nat \u223c A\n\ndef ctble {U : Type} (A : Set U) : Prop :=\n finite A \u2228 denum A\n\ndef numElts {U : Type} (A : Set U) (n : Nat) : Prop := I n \u223c A\n\ndef emptyRel (U V : Type) (x : U) (y : V) : Prop := False\n\ndef remove_one {U V : Type} (R : Rel U V) (u : U) (v : V)\n (x : U) (y : V) : Prop := x \u2260 u \u2227 y \u2260 v \u2227 (R x y \u2228 (R x v \u2227 R u y))\n\ndef one_match {U V : Type} (a : U) (b : V)\n (x : U) (y : V) : Prop := x = a \u2227 y = b\n\n/- Section 8.1 -/\nlemma invRel_def {U V : Type} (R : Rel U V) (u : U) (v : V) :\n invRel R v u \u2194 R u v := by rfl\n\ntheorem equinum_image {U V : Type} {A : Set U} {B : Set V} {f : U \u2192 V}\n (h1 : one_one_on f A) (h2 : image f A = B) : A \u223c B := by\n rewrite [\u2190h2]\n define --Goal : \u2203 (R : Rel U V), matching R A (image f A)\n set R : Rel U V := RelWithinFromFunc f A\n apply Exists.intro R\n define --Goal : rel_within R A (image f A) \u2227\n --fcnl_on R A \u2227 fcnl_on (invRel R) (image f A)\n apply And.intro\n \u00b7 -- Proof of rel_within\n define --Goal : \u2200 \u2983x : U\u2984 \u2983y : V\u2984, R x y \u2192 x \u2208 A \u2227 y \u2208 image f A\n fix x : U; fix y : V\n assume h3 : R x y --Goal : x \u2208 A \u2227 y \u2208 image f A\n define at h3 --h3 : x \u2208 A \u2227 f x = y\n apply And.intro h3.left\n define\n show \u2203 x \u2208 A, f x = y from Exists.intro x h3\n done\n \u00b7 -- Proofs of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on R A\n define --Goal : \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203! (y : V), R x y\n fix x : U\n assume h3 : x \u2208 A\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (f x)\n define --Goal : x \u2208 A \u2227 f x = f x\n apply And.intro h3\n rfl\n done\n \u00b7 -- Uniqueness\n fix y1 : V; fix y2 : V\n assume h4 : R x y1\n assume h5 : R x y2 --Goal : y1 = y2\n define at h4; define at h5\n --h4 : x \u2208 A \u2227 f x = y1; h5 : x \u2208 A \u2227 f x = y2\n rewrite [h4.right] at h5\n show y1 = y2 from h5.right\n done\n done\n \u00b7 -- Proof of fcnl_on (invRel R) (image f A)\n define --Goal : \u2200 \u2983x : V\u2984, x \u2208 image f A \u2192 \u2203! (y : U), invRel R x y\n fix y : V\n assume h3 : y \u2208 image f A\n obtain (x : U) (h4 : x \u2208 A \u2227 f x = y) from h3\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro x\n define\n show x \u2208 A \u2227 f x = y from h4\n done\n \u00b7 -- Uniqueness\n fix x1 : U; fix x2 : U\n assume h5 : invRel R y x1\n assume h6 : invRel R y x2\n define at h5; define at h6\n --h5 : x1 \u2208 A \u2227 f x1 = y; h6 : x2 \u2208 A \u2227 f x2 = y\n rewrite [\u2190h6.right] at h5\n show x1 = x2 from h1 h5.left h6.left h5.right\n done\n done\n done\n done\n\nlemma id_one_one_on {U : Type} (A : Set U) : one_one_on id A := by\n define\n fix x1 : U; fix x2 : U\n assume h1 : x1 \u2208 A\n assume h2 : x2 \u2208 A\n assume h3 : id x1 = id x2\n show x1 = x2 from h3\n done\n\nlemma image_id {U : Type} (A : Set U) : image id A = A := by\n apply Set.ext\n fix x : U\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : x \u2208 image id A\n obtain (y : U) (h2 : y \u2208 A \u2227 id y = x) from h1\n rewrite [\u2190h2.right]\n show id y \u2208 A from h2.left\n done\n \u00b7 -- (\u2190)\n assume h1 : x \u2208 A\n apply Exists.intro x --Goal : x \u2208 A \u2227 id x = x\n apply And.intro h1\n rfl\n done\n done\n\ntheorem Theorem_8_1_3_1 {U : Type} (A : Set U) : A \u223c A :=\n equinum_image (id_one_one_on A) (image_id A)\n\nlemma inv_inv {U V : Type} (R : Rel U V) : invRel (invRel R) = R := by rfl\n\nlemma inv_match {U V : Type} {R : Rel U V} {A : Set U} {B : Set V}\n (h : matching R A B) : matching (invRel R) B A := by\n define --Goal : rel_within (invRel R) B A \u2227\n --fcnl_on (invRel R) B \u2227 fcnl_on (invRel (invRel R)) A\n define at h --h : rel_within R A B \u2227 fcnl_on R A \u2227 fcnl_on (invRel R) B\n apply And.intro\n \u00b7 -- Proof that rel_within (invRel R) B A\n define --Goal : \u2200 \u2983x : V\u2984 \u2983y : U\u2984, invRel R x y \u2192 x \u2208 B \u2227 y \u2208 A\n fix y : V; fix x : U\n assume h1 : invRel R y x\n define at h1 --h1 : R x y\n have h2 : x \u2208 A \u2227 y \u2208 B := h.left h1\n show y \u2208 B \u2227 x \u2208 A from And.intro h2.right h2.left\n done\n \u00b7 -- proof that fcnl_on (inv R) B \u2227 fcnl_on (inv (inv R)) A\n rewrite [inv_inv]\n show fcnl_on (invRel R) B \u2227 fcnl_on R A from\n And.intro h.right.right h.right.left\n done\n done\n\ntheorem Theorem_8_1_3_2 {U V : Type} {A : Set U} {B : Set V}\n (h : A \u223c B) : B \u223c A := by\n obtain (R : Rel U V) (h1 : matching R A B) from h\n apply Exists.intro (invRel R)\n show matching (invRel R) B A from inv_match h1\n done\n\nlemma fcnl_exists {U V : Type} {R : Rel U V} {A : Set U} {x : U}\n (h1 : fcnl_on R A) (h2 : x \u2208 A) : \u2203 (y : V), R x y := by\n define at h1\n obtain (y : V) (h3 : R x y)\n (h4 : \u2200 (y_1 y_2 : V), R x y_1 \u2192 R x y_2 \u2192 y_1 = y_2) from h1 h2\n show \u2203 (y : V), R x y from Exists.intro y h3\n done\n\nlemma fcnl_unique {U V : Type}\n {R : Rel U V} {A : Set U} {x : U} {y1 y2 : V} (h1 : fcnl_on R A)\n (h2 : x \u2208 A) (h3 : R x y1) (h4 : R x y2) : y1 = y2 := by\n define at h1\n obtain (z : V) (h5 : R x z)\n (h6 : \u2200 (y_1 y_2 : V), R x y_1 \u2192 R x y_2 \u2192 y_1 = y_2) from h1 h2\n show y1 = y2 from h6 y1 y2 h3 h4\n done\n\nlemma compRel_def {U V W : Type}\n (S : Rel V W) (R : Rel U V) (u : U) (w : W) :\n compRel S R u w \u2194 \u2203 (x : V), R u x \u2227 S x w := by rfl\n\nlemma inv_comp {U V W : Type} (R : Rel U V) (S : Rel V W) :\n invRel (compRel S R) = compRel (invRel R) (invRel S) :=\n calc invRel (compRel S R)\n _ = RelFromExt (inv (comp (extension S) (extension R))) := by rfl\n _ = RelFromExt (comp (inv (extension R)) (inv (extension S))) := by\n rw [Theorem_4_2_5_5]\n _ = compRel (invRel R) (invRel S) := by rfl\n\nlemma comp_fcnl {U V W : Type} {R : Rel U V} {S : Rel V W}\n {A : Set U} {B : Set V} {C : Set W} (h1 : matching R A B)\n (h2 : matching S B C) : fcnl_on (compRel S R) A := by\n define; define at h1; define at h2\n fix a : U\n assume h3 : a \u2208 A\n obtain (b : V) (h4 : R a b) from fcnl_exists h1.right.left h3\n have h5 : a \u2208 A \u2227 b \u2208 B := h1.left h4\n obtain (c : W) (h6 : S b c) from fcnl_exists h2.right.left h5.right\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro c\n rewrite [compRel_def]\n show \u2203 (x : V), R a x \u2227 S x c from Exists.intro b (And.intro h4 h6)\n done\n \u00b7 -- Uniqueness\n fix c1 : W; fix c2 : W\n assume h7 : compRel S R a c1\n assume h8 : compRel S R a c2 --Goal : c1 = c2\n rewrite [compRel_def] at h7\n rewrite [compRel_def] at h8\n obtain (b1 : V) (h9 : R a b1 \u2227 S b1 c1) from h7\n obtain (b2 : V) (h10 : R a b2 \u2227 S b2 c2) from h8\n have h11 : b1 = b := fcnl_unique h1.right.left h3 h9.left h4\n have h12 : b2 = b := fcnl_unique h1.right.left h3 h10.left h4\n rewrite [h11] at h9\n rewrite [h12] at h10\n show c1 = c2 from\n fcnl_unique h2.right.left h5.right h9.right h10.right\n done\n done\n\nlemma comp_match {U V W : Type} {R : Rel U V} {S : Rel V W}\n {A : Set U} {B : Set V} {C : Set W} (h1 : matching R A B)\n (h2 : matching S B C) : matching (compRel S R) A C := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within (compRel S R) A C\n define\n fix a : U; fix c : W\n assume h3 : compRel S R a c\n rewrite [compRel_def] at h3\n obtain (b : V) (h4 : R a b \u2227 S b c) from h3\n have h5 : a \u2208 A \u2227 b \u2208 B := h1.left h4.left\n have h6 : b \u2208 B \u2227 c \u2208 C := h2.left h4.right\n show a \u2208 A \u2227 c \u2208 C from And.intro h5.left h6.right\n done\n \u00b7 -- Proof of fcnl_on statements\n apply And.intro\n \u00b7 -- Proof of fcnl_on (compRel S R) A\n show fcnl_on (compRel S R) A from comp_fcnl h1 h2\n done\n \u00b7 -- Proof of fcnl_on (invRel (compRel S R)) Z\n rewrite [inv_comp]\n have h3 : matching (invRel R) B A := inv_match h1\n have h4 : matching (invRel S) C B := inv_match h2\n show fcnl_on (compRel (invRel R) (invRel S)) C from comp_fcnl h4 h3\n done\n done\n done\n\ntheorem Theorem_8_1_3_3 {U V W : Type} {A : Set U} {B : Set V} {C : Set W}\n (h1 : A \u223c B) (h2 : B \u223c C) : A \u223c C := by\n obtain (R : Rel U V) (h3 : matching R A B) from h1\n obtain (S : Rel V W) (h4 : matching S B C) from h2\n apply Exists.intro (compRel S R)\n show matching (compRel S R) A C from comp_match h3 h4\n done\n\nlemma I_def (k n : Nat) : k \u2208 I n \u2194 k < n := by rfl\n\nlemma denum_def {U : Type} (A : Set U) : denum A \u2194 Univ Nat \u223c A := by rfl\n\n/- Section 8.1\u00bd -/\nlemma numElts_def {U : Type} (A : Set U) (n : Nat) :\n numElts A n \u2194 I n \u223c A := by rfl\n\nlemma finite_def {U : Type} (A : Set U) :\n finite A \u2194 \u2203 (n : Nat), numElts A n := by rfl\n\nlemma fcnl_on_empty {U V : Type}\n (R : Rel U V) {A : Set U} (h1 : empty A) : fcnl_on R A := by\n define\n fix a : U\n assume h2 : a \u2208 A --Goal : \u2203! (y : V), R a y\n contradict h1 with h3 --Goal : \u2203 (x : U), x \u2208 A\n show \u2203 (x : U), x \u2208 A from Exists.intro a h2\n done\n\n", "theoremStatement": "lemma empty_match {U V : Type} {A : Set U} {B : Set V}\n (h1 : empty A) (h2 : empty B) : matching (emptyRel U V) A B ", "theoremName": "HTPI.empty_match", "fileCreated": {"commit": "16a90bdd1803de01730cb7eb7cfdf6ace669bfbf", "date": "2023-03-26"}, "theoremCreated": {"commit": "8ac9101157d92dfd028ad1068e8b4cac2f343eaf", "date": "2023-01-31"}, "file": "htpi/HTPILib/Chap8Part1.lean", "module": "HTPILib.Chap8Part1", "jsonFile": "HTPILib.Chap8Part1.jsonl", "positionMetadata": {"lineInFile": 303, "tokenPositionInFile": 9921, "theoremPositionInFile": 38}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 6, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 24, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", 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"Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix a : U; fix b : V\n assume h3 : emptyRel U V a b --Goal : a \u2208 A \u2227 b \u2208 B\n by_contra h4 --Goal : False\n define at h3\n show False from h3\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on emptyRel\n show fcnl_on (emptyRel U V) A from fcnl_on_empty (emptyRel U V) h1\n done\n \u00b7 -- Proof of fcnl_on (invRel emptyRel)\n show fcnl_on (invRel (emptyRel U V)) B from\n fcnl_on_empty (invRel (emptyRel U V)) h2\n done\n done", "proofType": "tactic", "proofLengthLines": 20, "proofLengthTokens": 577}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\n", "theoremStatement": "lemma dvd_self (n : Nat) : n \u2223 n ", "theoremName": "HTPI.dvd_self", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 152, "tokenPositionInFile": 4240, "theoremPositionInFile": 32}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, 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"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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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", 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Velleman -/\n\nimport HTPILib.Chap4\nnamespace HTPI\n\n/- Definitions -/\ndef graph {A B : Type} (f : A \u2192 B) : Set (A \u00d7 B) :=\n {(a, b) : A \u00d7 B | f a = b}\n\ndef is_func_graph {A B : Type} (G : Set (A \u00d7 B)) : Prop :=\n \u2200 (x : A), \u2203! (y : B), (x, y) \u2208 G\n\ndef onto {A B : Type} (f : A \u2192 B) : Prop :=\n \u2200 (y : B), \u2203 (x : A), f x = y\n\ndef one_to_one {A B : Type} (f : A \u2192 B) : Prop :=\n \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n\ndef closed {A : Type} (f : A \u2192 A) (C : Set A) : Prop := \u2200 x \u2208 C, f x \u2208 C\n\ndef closure {A : Type} (f : A \u2192 A) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed f D}\n\ndef closed2 {A : Type} (f : A \u2192 A \u2192 A) (C : Set A) : Prop :=\n \u2200 x \u2208 C, \u2200 y \u2208 C, f x y \u2208 C\n\ndef closure2 {A : Type} (f : A \u2192 A \u2192 A) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed2 f D}\n\ndef closed_family {A : Type} (F : Set (A \u2192 A)) (C : Set A) : Prop :=\n \u2200 f \u2208 F, closed f C\n\ndef closure_family {A : Type} (F : Set (A \u2192 A)) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed_family F D}\n\ndef image {A B : Type} (f : A \u2192 B) (X : Set A) : Set B :=\n {f x | x \u2208 X}\n\ndef inverse_image {A B : Type} (f : A \u2192 B) (Y : Set B) : Set A :=\n {a : A | f a \u2208 Y}\n\n/- Section 5.1 -/\ntheorem graph_def {A B : Type} (f : A \u2192 B) (a : A) (b : B) :\n (a, b) \u2208 graph f \u2194 f a = b := by rfl\n\ntheorem func_from_graph_ltr {A B : Type} (F : Set (A \u00d7 B)) :\n (\u2203 (f : A \u2192 B), graph f = F) \u2192 is_func_graph F := sorry\n\n--This proof is explained in Section 8.2\ntheorem func_from_graph_rtl {A B : Type} (F : Set (A \u00d7 B)) :\n is_func_graph F \u2192 (\u2203 (f : A \u2192 B), graph f = F) := by\n assume h1 : is_func_graph F\n define at h1 --h1 : \u2200 (x : A), \u2203! (y : B), (x, y) \u2208 F\n have h2 : \u2200 (x : A), \u2203 (y : B), (x, y) \u2208 F := by\n fix x : A\n obtain (y : B) (h3 : (x, y) \u2208 F)\n (h4 : \u2200 (y1 y2 : B), (x, y1) \u2208 F \u2192 (x, y2) \u2208 F \u2192 y1 = y2) from h1 x\n show \u2203 (y : B), (x, y) \u2208 F from Exists.intro y h3\n done\n set f : A \u2192 B := fun (x : A) => Classical.choose (h2 x)\n apply Exists.intro f\n apply Set.ext\n fix (x, y) : A \u00d7 B\n have h3 : (x, f x) \u2208 F := Classical.choose_spec (h2 x)\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h4 : (x, y) \u2208 graph f\n define at h4 --h4 : f x = y\n rewrite [h4] at h3\n show (x, y) \u2208 F from h3\n done\n \u00b7 -- (\u2190)\n assume h4 : (x, y) \u2208 F\n define --Goal : f x = y\n obtain (z : B) (h5 : (x, z) \u2208 F)\n (h6 : \u2200 (y1 y2 : B), (x, y1) \u2208 F \u2192 (x, y2) \u2208 F \u2192 y1 = y2) from h1 x\n show f x = y from h6 (f x) y h3 h4\n done\n done\n\ntheorem func_from_graph {A B : Type} (F : Set (A \u00d7 B)) :\n (\u2203 (f : A \u2192 B), graph f = F) \u2194 is_func_graph F :=\n Iff.intro (func_from_graph_ltr F) (func_from_graph_rtl F)\n\ntheorem Theorem_5_1_4 {A B : Type} (f g : A \u2192 B) :\n (\u2200 (a : A), f a = g a) \u2192 f = g := funext\n\nexample {A B : Type} (f g : A \u2192 B) :\n graph f = graph g \u2192 f = g := by\n assume h1 : graph f = graph g --Goal : f = g\n apply funext --Goal : \u2200 (x : A), f x = g x\n fix x : A\n have h2 : (x, f x) \u2208 graph f := by\n define --Goal : f x = f x\n rfl\n done\n rewrite [h1] at h2 --h2 : (x, f x) \u2208 graph g\n define at h2 --h2 : g x = f x\n show f x = g x from h2.symm\n done\n\ndef square1 (n : Nat) : Nat := n ^ 2\n\ndef square2 : Nat \u2192 Nat := fun (n : Nat) => n ^ 2\n\nexample : square1 = square2 := by rfl\n\n#eval square1 7 --Answer: 49\n\ntheorem Theorem_5_1_5 {A B C : Type} (f : A \u2192 B) (g : B \u2192 C) :\n \u2203 (h : A \u2192 C), graph h = comp (graph g) (graph f) := by\n set h : A \u2192 C := fun (x : A) => g (f x)\n apply Exists.intro h\n apply Set.ext\n fix (a, c) : A \u00d7 C\n apply Iff.intro\n \u00b7 -- Proof that (a, c) \u2208 graph h \u2192 (a, c) \u2208 comp (graph g) (graph f)\n assume h1 : (a, c) \u2208 graph h\n define at h1 --h1 : h a = c\n define --Goal : \u2203 (x : B), (a, x) \u2208 graph f \u2227 (x, c) \u2208 graph g\n apply Exists.intro (f a)\n apply And.intro\n \u00b7 -- Proof that (a, f a) \u2208 graph f\n define\n rfl\n done\n \u00b7 -- Proof that (f a, c) \u2208 graph g\n define\n show g (f a) = c from h1\n done\n done\n \u00b7 -- Proof that (a, c) \u2208 comp (graph g) (graph f) \u2192 (a, c) \u2208 graph h\n assume h1 : (a, c) \u2208 comp (graph g) (graph f)\n define --Goal : h a = c\n define at h1 --h1 : \u2203 (x : B), (a, x) \u2208 graph f \u2227 (x, c) \u2208 graph g\n obtain (b : B) (h2 : (a, b) \u2208 graph f \u2227 (b, c) \u2208 graph g) from h1\n have h3 : (a, b) \u2208 graph f := h2.left\n have h4 : (b, c) \u2208 graph g := h2.right\n define at h3 --h3 : f a = b\n define at h4 --h4 : g b = c\n rewrite [\u2190h3] at h4 --h4 : g (f a) = c\n show h a = c from h4\n done\n done\n\nexample {A B C D : Type} (f : A \u2192 B) (g : B \u2192 C) (h : C \u2192 D) :\n h \u2218 (g \u2218 f) = (h \u2218 g) \u2218 f := by rfl\n\nexample {A B : Type} (f : A \u2192 B) : f \u2218 id = f := by rfl\n\nexample {A B : Type} (f : A \u2192 B) : id \u2218 f = f := by rfl\n\n/- Section 5.2 -/\n", "theoremStatement": "theorem Theorem_5_2_5_1 {A B C : Type} (f : A \u2192 B) (g : B \u2192 C) :\n one_to_one f \u2192 one_to_one g \u2192 one_to_one (g \u2218 f) ", "theoremName": "HTPI.Theorem_5_2_5_1", "fileCreated": {"commit": "cc48e6ba4c656bf74f7690323b2c27f2c466665b", "date": "2023-03-11"}, "theoremCreated": {"commit": "190e34021c4943fc63cee05edf567875671378bb", "date": "2023-02-18"}, "file": "htpi/HTPILib/Chap5.lean", "module": "HTPILib.Chap5", "jsonFile": "HTPILib.Chap5.jsonl", "positionMetadata": {"lineInFile": 154, "tokenPositionInFile": 4901, "theoremPositionInFile": 20}, 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"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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n assume h1 : one_to_one f\n assume h2 : one_to_one g\n define at h1 --h1 : \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n define at h2 --h2 : \u2200 (x1 x2 : B), g x1 = g x2 \u2192 x1 = x2\n define --Goal : \u2200 (x1 x2 : A), (g \u2218 f) x1 = (g \u2218 f) x2 \u2192 x1 = x2\n fix a1 : A\n fix a2 : A --Goal : (g \u2218 f) a1 = (g \u2218 f) a2 \u2192 a1 = a2\n define : (g \u2218 f) a1; define : (g \u2218 f) a2\n --Goal : g (f a1) = g (f a2) \u2192 a1 = a2\n assume h3 : g (f a1) = g (f a2)\n have h4 : f a1 = f a2 := h2 (f a1) (f a2) h3\n show a1 = a2 from h1 a1 a2 h4\n done", "proofType": "tactic", "proofLengthLines": 13, "proofLengthTokens": 543}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\nlemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) := by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done\n\nlemma F_invertible (m i : Nat) : invertible (F m i) := by\n by_cases h : rel_prime m i\n \u00b7 -- Case 1. h : rel_prime m i\n rewrite [F_rp_def h]\n show invertible [i]_m from (Theorem_7_3_7 m i).rtl h\n done\n \u00b7 -- Case 2. h : \u00acrel_prime m i\n rewrite [F_not_rp_def h]\n apply Exists.intro [1]_m\n show [1]_m * [1]_m = [1]_m from Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma Fprod_invertible (m : Nat) :\n \u2200 (k : Nat), invertible (prod_seq k 0 (F m)) := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro [1]_m\n show prod_seq 0 0 (F m) * [1]_m = [1]_m from\n calc prod_seq 0 0 (F m) * [1]_m\n _ = [1]_m * [1]_m := by rw [prod_seq_base]\n _ = [1]_m := Theorem_7_3_6_7 ([1]_m)\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : invertible (prod_seq k 0 (F m))\n rewrite [prod_seq_zero_step]\n show invertible (prod_seq k 0 (F m) * (F m k)) from\n (prod_inv_iff_inv ih (F m k)).rtl (F_invertible m k)\n done\n done\n\ntheorem Theorem_7_4_2 {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m ^ (phi m) = [1]_m := by\n have h2 : invertible (prod_seq m 0 (F m)) := Fprod_invertible m m\n obtain (Y : ZMod m) (h3 : prod_seq m 0 (F m) * Y = [1]_m) from h2\n show [a]_m ^ (phi m) = [1]_m from\n calc [a]_m ^ (phi m)\n _ = [a]_m ^ (phi m) * [1]_m := (Theorem_7_3_6_7 _).symm\n _ = [a]_m ^ (phi m) * (prod_seq m 0 (F m) * Y) := by rw [h3]\n _ = ([a]_m ^ (phi m) * prod_seq m 0 (F m)) * Y := by ring\n _ = prod_seq m 0 (F m \u2218 G m a) * Y := by rw [FG_prod h1 m, phi_def]\n _ = prod_seq m 0 (F m) * Y := by\n rw [perm_prod (F m) m (G m a) (G_perm_below h1)]\n _ = [1]_m := by rw [h3]\n done\n\nlemma Exercise_7_4_5_Int (m : Nat) (a : Int) :\n \u2200 (n : Nat), [a]_m ^ n = [a ^ n]_m := sorry\n\n", "theoremStatement": "lemma Exercise_7_4_5_Nat (m a n : Nat) :\n [a]_m ^ n = [a ^ n]_m ", "theoremName": "HTPI.Exercise_7_4_5_Nat", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "7c1bd1c8e2a635b582a13808075ad51997f1e19f", "date": "2023-04-21"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1713, "tokenPositionInFile": 56043, "theoremPositionInFile": 156}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, 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"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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rewrite [Exercise_7_4_5_Int]\n rfl\n done", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 49}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\n", "theoremStatement": "lemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n ", "theoremName": "HTPI.prod_seq_zero_step", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "515de92e6c12bba8178aad612650f80eacd0a245", "date": "2023-04-13"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1159, "tokenPositionInFile": 36467, "theoremPositionInFile": 110}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 27, "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.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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rewrite [prod_seq_step, zero_add]\n rfl\n done", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 54}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\n", "theoremStatement": "theorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b ", "theoremName": "HTPI.Theorem_7_2_2", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": 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"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.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.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "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.Meta.Tactic.Delta", "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.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "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.Meta.Tactic.Symm", "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.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "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", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done", "proofType": "tactic", "proofLengthLines": 17, "proofLengthTokens": 717}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\nlemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f := by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done\n\nlemma break_prod_twice {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by\n have h2 : prod_seq (n + 1) 0 f = prod_seq n 0 f * prod_seq 1 n f :=\n break_prod n f 1\n rewrite [prod_one] at h2\n have h3 : prod_seq (u + 1 + j) 0 f = prod_seq (u + 1) 0 f * prod_seq j (u + 1) f :=\n break_prod (u + 1) f j\n rewrite [\u2190h1] at h3\n have h4 : prod_seq (u + 1) 0 f = prod_seq u 0 f * prod_seq 1 u f :=\n break_prod u f 1\n rewrite [prod_one] at h4\n rewrite [h3, h4] at h2\n show prod_seq (n + 1) 0 f = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n from h2\n done\n\nlemma swap_prod_eq_prod {m u n : Nat} (f : Nat \u2192 ZMod m) (h1 : u \u2264 n) :\n prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f := by\n by_cases h2 : u = n\n \u00b7 -- Case 1. h2 : u = n\n rewrite [h2, trivial_swap n]\n --Goal : prod_seq (n + 1) 0 (f \u2218 id) = prod_seq (n + 1) 0 f\n rfl\n done\n \u00b7 -- Case 2. h2 : \u00acu = n\n have h3 : u + 1 \u2264 n := Nat.lt_of_le_of_ne h1 h2\n obtain (j : Nat) (h4 : n = u + 1 + j) from Nat.exists_eq_add_of_le h3\n have break_f : prod_seq (n + 1) 0 f =\n prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n :=\n break_prod_twice f h4\n have break_fs : prod_seq (n + 1) 0 (f \u2218 swap u n) =\n prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_prod_twice (f \u2218 swap u n) h4\n have f_eq_fs_below : prod_seq u 0 (f \u2218 swap u n) =\n prod_seq u 0 f := swap_prod_eq_prod_below f h1\n have f_eq_fs_btwn : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := swap_prod_eq_prod_between f h4\n show prod_seq (n + 1) 0 (f \u2218 swap u n) = prod_seq (n + 1) 0 f from\n calc prod_seq (n + 1) 0 (f \u2218 swap u n)\n _ = prod_seq u 0 (f \u2218 swap u n) * (f \u2218 swap u n) u *\n prod_seq j (u + 1) (f \u2218 swap u n) * (f \u2218 swap u n) n :=\n break_fs\n _ = prod_seq u 0 f * (f \u2218 swap u n) u *\n prod_seq j (u + 1) f * (f \u2218 swap u n) n := by\n rw [f_eq_fs_below, f_eq_fs_btwn]\n _ = prod_seq u 0 f * f (swap u n u) *\n prod_seq j (u + 1) f * f (swap u n n) := by rfl\n _ = prod_seq u 0 f * f n * prod_seq j (u + 1) f * f u := by\n rw [swap_fst, swap_snd]\n _ = prod_seq u 0 f * f u * prod_seq j (u + 1) f * f n := by ring\n _ = prod_seq (n + 1) 0 f := break_f.symm\n done\n done\n\nlemma perm_below_fixed {n : Nat} {g : Nat \u2192 Nat}\n (h1 : perm_below (n + 1) g) (h2 : g n = n) : perm_below n g := sorry\n\n", "theoremStatement": "lemma perm_prod {m : Nat} (f : Nat \u2192 ZMod m) :\n \u2200 (n : Nat), \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g) ", "theoremName": "HTPI.perm_prod", "fileCreated": 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"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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n by_induc\n \u00b7 -- Base Case\n fix g : Nat \u2192 Nat\n assume h1 : perm_below 0 g\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (g : Nat \u2192 Nat), perm_below n g \u2192\n prod_seq n 0 f = prod_seq n 0 (f \u2218 g)\n fix g : Nat \u2192 Nat\n assume g_pb : perm_below (n + 1) g\n define at g_pb\n have g_ob : onto_below (n + 1) g := g_pb.right.right\n define at g_ob\n have h1 : n < n + 1 := by linarith\n obtain (u : Nat) (h2 : u < n + 1 \u2227 g u = n) from g_ob n h1\n have s_pb : perm_below (n + 1) (swap u n) :=\n swap_perm_below h2.left h1\n have gs_pb_n1 : perm_below (n + 1) (g \u2218 swap u n) :=\n comp_perm_below g_pb s_pb\n have gs_fix_n : (g \u2218 swap u n) n = n :=\n calc (g \u2218 swap u n) n\n _ = g (swap u n n) := by rfl\n _ = g u := by rw [swap_snd]\n _ = n := h2.right\n have gs_pb_n : perm_below n (g \u2218 swap u n) :=\n perm_below_fixed gs_pb_n1 gs_fix_n\n have gs_prod : prod_seq n 0 f = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) :=\n ih (g \u2218 swap u n) gs_pb_n\n have h3 : u \u2264 n := by linarith\n show prod_seq (n + 1) 0 f = prod_seq (n + 1) 0 (f \u2218 g) from\n calc prod_seq (n + 1) 0 f\n _ = prod_seq n 0 f * f n := prod_seq_zero_step n f\n _ = prod_seq n 0 (f \u2218 (g \u2218 swap u n)) *\n f ((g \u2218 swap u n) n) := by rw [gs_prod, gs_fix_n]\n _ = prod_seq n 0 (f \u2218 g \u2218 swap u n) *\n (f \u2218 g \u2218 swap u n) n := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g \u2218 swap u n) :=\n (prod_seq_zero_step n (f \u2218 g \u2218 swap u n)).symm\n _ = prod_seq (n + 1) 0 ((f \u2218 g) \u2218 swap u n) := by rfl\n _ = prod_seq (n + 1) 0 (f \u2218 g) := swap_prod_eq_prod (f \u2218 g) h3\n done\n done", "proofType": "tactic", "proofLengthLines": 45, "proofLengthTokens": 1728}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\n", "theoremStatement": "lemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m ", "theoremName": "HTPI.cc_neg_zero_iff_cc_zero", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 893, "tokenPositionInFile": 27651, "theoremPositionInFile": 85}, "dependencyMetadata": {"inFilePremises": true, 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"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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 239}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap3\nnamespace HTPI\n\n/- Definitions -/\ndef Dom {A B : Type} (R : Set (A \u00d7 B)) : Set A :=\n {a : A | \u2203 (b : B), (a, b) \u2208 R}\n\ndef Ran {A B : Type} (R : Set (A \u00d7 B)) : Set B :=\n {b : B | \u2203 (a : A), (a, b) \u2208 R}\n\ndef inv {A B : Type} (R : Set (A \u00d7 B)) : Set (B \u00d7 A) :=\n {(b, a) : B \u00d7 A | (a, b) \u2208 R}\n\ndef comp {A B C : Type}\n (S : Set (B \u00d7 C)) (R : Set (A \u00d7 B)) : Set (A \u00d7 C) :=\n {(a, c) : A \u00d7 C | \u2203 (x : B), (a, x) \u2208 R \u2227 (x, c) \u2208 S}\n\ndef extension {A B : Type} (R : Rel A B) : Set (A \u00d7 B) :=\n {(a, b) : A \u00d7 B | R a b}\n\ndef reflexive {A : Type} (R : BinRel A) : Prop :=\n \u2200 (x : A), R x x\n\ndef symmetric {A : Type} (R : BinRel A) : Prop :=\n \u2200 (x y : A), R x y \u2192 R y x\n\ndef transitive {A : Type} (R : BinRel A) : Prop :=\n \u2200 (x y z : A), R x y \u2192 R y z \u2192 R x z\n\ndef elementhood (A : Type) (a : A) (X : Set A) : Prop := a \u2208 X\n\ndef RelFromExt {A B : Type}\n (R : Set (A \u00d7 B)) (a : A) (b : B) : Prop := (a, b) \u2208 R\n\ndef antisymmetric {A : Type} (R : BinRel A) : Prop :=\n \u2200 (x y : A), R x y \u2192 R y x \u2192 x = y\n\ndef partial_order {A : Type} (R : BinRel A) : Prop :=\n reflexive R \u2227 transitive R \u2227 antisymmetric R\n\ndef total_order {A : Type} (R : BinRel A) : Prop :=\n partial_order R \u2227 \u2200 (x y : A), R x y \u2228 R y x\n\ndef sub (A : Type) (X Y : Set A) : Prop := X \u2286 Y\n\ndef smallestElt {A : Type} (R : BinRel A) (b : A) (B : Set A) : Prop :=\n b \u2208 B \u2227 \u2200 x \u2208 B, R b x\n\ndef minimalElt {A : Type} (R : BinRel A) (b : A) (B : Set A) : Prop :=\n b \u2208 B \u2227 \u00ac\u2203 x \u2208 B, R x b \u2227 x \u2260 b\n\ndef upperBd {A : Type} (R : BinRel A) (a : A) (B : Set A) : Prop :=\n \u2200 x \u2208 B, R x a\n\ndef lub {A : Type} (R : BinRel A) (a : A) (B : Set A) : Prop :=\n smallestElt R a {c : A | upperBd R c B}\n\ndef equiv_rel {A : Type} (R : BinRel A) : Prop :=\n reflexive R \u2227 symmetric R \u2227 transitive R\n\ndef equivClass {A : Type} (R : BinRel A) (x : A) : Set A :=\n {y : A | R y x}\n\ndef mod (A : Type) (R : BinRel A) : Set (Set A) :=\n {equivClass R x | x : A}\n\ndef empty {A : Type} (X : Set A) : Prop := \u00ac\u2203 (x : A), x \u2208 X\n\ndef pairwise_disjoint {A : Type} (F : Set (Set A)) : Prop :=\n \u2200 X \u2208 F, \u2200 Y \u2208 F, X \u2260 Y \u2192 empty (X \u2229 Y)\n\ndef partition {A : Type} (F : Set (Set A)) : Prop :=\n (\u2200 (x : A), x \u2208 \u22c3\u2080 F) \u2227 pairwise_disjoint F \u2227 \u2200 X \u2208 F, \u00acempty X\n\ndef EqRelFromPart {A : Type} (F : Set (Set A)) (x y : A) : Prop :=\n \u2203 X \u2208 F, x \u2208 X \u2227 y \u2208 X\n\n/- Section 4.2 -/\ntheorem Theorem_4_2_5_1 {A B : Type}\n (R : Set (A \u00d7 B)) : inv (inv R) = R := by rfl\n\ntheorem Theorem_4_2_5_2 {A B : Type}\n (R : Set (A \u00d7 B)) : Dom (inv R) = Ran R := by rfl\n\ntheorem Theorem_4_2_5_3 {A B : Type}\n (R : Set (A \u00d7 B)) : Ran (inv R) = Dom R := by rfl\n\ntheorem Theorem_4_2_5_4 {A B C D : Type}\n (R : Set (A \u00d7 B)) (S : Set (B \u00d7 C)) (T : Set (C \u00d7 D)) :\n comp T (comp S R) = comp (comp T S) R := by\n apply Set.ext\n fix (a, d) : A \u00d7 D\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : (a, d) \u2208 comp T (comp S R)\n --Goal : (a, d) \u2208 comp (comp T S) R\n define --Goal : \u2203 (x : B), (a, x) \u2208 R \u2227 (x, d) \u2208 comp T S\n define at h1 --h1 : \u2203 (x : C), (a, x) \u2208 comp S R \u2227 (x, d) \u2208 T\n obtain (c : C) (h2 : (a, c) \u2208 comp S R \u2227 (c, d) \u2208 T) from h1\n have h3 : (a, c) \u2208 comp S R := h2.left\n define at h3 --h3 : \u2203 (x : B), (a, x) \u2208 R \u2227 (x, c) \u2208 S\n obtain (b : B) (h4 : (a, b) \u2208 R \u2227 (b, c) \u2208 S) from h3\n apply Exists.intro b --Goal : (a, b) \u2208 R \u2227 (b, d) \u2208 comp T S\n apply And.intro h4.left --Goal : (b, d) \u2208 comp T S\n define --Goal : \u2203 (x : C), (b, x) \u2208 S \u2227 (x, d) \u2208 T\n show \u2203 (x : C), (b, x) \u2208 S \u2227 (x, d) \u2208 T from\n Exists.intro c (And.intro h4.right h2.right)\n done\n \u00b7 -- (\u2190)\n assume h1 : (a, d) \u2208 comp (comp T S) R\n define; define at h1\n obtain (b : B) (h2 : (a, b) \u2208 R \u2227 (b, d) \u2208 comp T S) from h1\n have h3 : (b, d) \u2208 comp T S := h2.right\n define at h3\n obtain (c : C) (h4 : (b, c) \u2208 S \u2227 (c, d) \u2208 T) from h3\n apply Exists.intro c\n apply And.intro _ h4.right\n define\n show \u2203 (x : B), (a, x) \u2208 R \u2227 (x, c) \u2208 S from\n Exists.intro b (And.intro h2.left h4.left)\n done\n done\n\ntheorem inv_def {A B : Type} (R : Set (A \u00d7 B)) (a : A) (b : B) :\n (b, a) \u2208 inv R \u2194 (a, b) \u2208 R := by rfl\n\ntheorem Theorem_4_2_5_5 {A B C : Type}\n (R : Set (A \u00d7 B)) (S : Set (B \u00d7 C)) :\n inv (comp S R) = comp (inv R) (inv S) := by\n apply Set.ext\n fix (c, a) : C \u00d7 A\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : (c, a) \u2208 inv (comp S R)\n --Goal : (c, a) \u2208 comp (inv R) (inv S)\n define at h1 --h1 : \u2203 (x : B), (a, x) \u2208 R \u2227 (x, c) \u2208 S\n define --Goal : \u2203 (x : B), (c, x) \u2208 inv S \u2227 (x, a) \u2208 inv R\n obtain (b : B) (h2 : (a, b) \u2208 R \u2227 (b, c) \u2208 S) from h1\n apply Exists.intro b --Goal : (c, b) \u2208 inv S \u2227 (b, a) \u2208 inv R\n rewrite [inv_def, inv_def] --Goal : (b, c) \u2208 S \u2227 (a, b) \u2208 R\n show (b, c) \u2208 S \u2227 (a, b) \u2208 R from And.intro h2.right h2.left\n done\n \u00b7 -- (\u2190)\n assume h1 : (c, a) \u2208 comp (inv R) (inv S)\n define at h1\n define\n obtain (b : B) (h2 : (c, b) \u2208 inv S \u2227 (b, a) \u2208 inv R) from h1\n apply Exists.intro b\n rewrite [inv_def, inv_def] at h2\n show (a, b) \u2208 R \u2227 (b, c) \u2208 S from And.intro h2.right h2.left\n done\n done\n\n/- Section 4.3 -/\ntheorem ext_def {A B : Type} (R : Rel A B) (a : A) (b : B) :\n (a, b) \u2208 extension R \u2194 R a b := by rfl\n\ntheorem Theorem_4_3_4_2 {A : Type} (R : BinRel A) :\n symmetric R \u2194 extension R = inv (extension R) := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : symmetric R\n define at h1 --h1 : \u2200 (x y : A), R x y \u2192 R y x\n apply Set.ext\n fix (a, b) : A \u00d7 A\n show (a, b) \u2208 extension R \u2194 (a, b) \u2208 inv (extension R) from\n calc (a, b) \u2208 extension R\n _ \u2194 R a b := by rfl\n _ \u2194 R b a := Iff.intro (h1 a b) (h1 b a)\n _ \u2194 (a, b) \u2208 inv (extension R) := by rfl\n done\n \u00b7 -- (\u2190)\n assume h1 : extension R = inv (extension R)\n define --Goal : \u2200 (x y : A), R x y \u2192 R y x\n fix a : A; fix b : A\n assume h2 : R a b --Goal : R b a\n rewrite [\u2190ext_def R, h1, inv_def, ext_def] at h2\n show R b a from h2\n done\n done\n\ntheorem RelFromExt_def {A B : Type}\n (R : Set (A \u00d7 B)) (a : A) (b : B) :\n RelFromExt R a b \u2194 (a, b) \u2208 R := by rfl\n\nexample {A B : Type} (R : Rel A B) :\n RelFromExt (extension R) = R := by rfl\n\nexample {A B : Type} (R : Set (A \u00d7 B)) :\n extension (RelFromExt R) = R := by rfl\n\n/- Section 4.4 -/\ntheorem Theorem_4_4_6_2 {A : Type} (R : BinRel A) (B : Set A) (b : A)\n (h1 : partial_order R) (h2 : smallestElt R b B) :\n minimalElt R b B \u2227 \u2200 (c : A), minimalElt R c B \u2192 b = c := by\n define at h1 --h1 : reflexive R \u2227 transitive R \u2227 antisymmetric R\n define at h2 --h2 : b \u2208 B \u2227 \u2200 (x : A), x \u2208 B \u2192 R b x\n apply And.intro\n \u00b7 -- Proof that b is minimal\n define --Goal : b \u2208 B \u2227 \u00ac\u2203 (x : A), x \u2208 B \u2227 R x b \u2227 x \u2260 b\n apply And.intro h2.left\n quant_neg --Goal : \u2200 (x : A), \u00ac(x \u2208 B \u2227 R x b \u2227 x \u2260 b)\n fix x : A\n demorgan --Goal : \u00acx \u2208 B \u2228 \u00ac(R x b \u2227 x \u2260 b)\n or_right with h3 --h3 : x \u2208 B; Goal : \u00ac(R x b \u2227 x \u2260 b)\n demorgan --Goal : \u00acR x b \u2228 x = b\n or_right with h4 --h4 : R x b; Goal : x = b\n have h5 : R b x := h2.right x h3\n have h6 : antisymmetric R := h1.right.right\n define at h6 --h6 : \u2200 (x y : A), R x y \u2192 R y x \u2192 x = y\n show x = b from h6 x b h4 h5\n done\n \u00b7 -- Proof that b is only minimal element\n fix c : A\n assume h3 : minimalElt R c B\n define at h3 --h3 : c \u2208 B \u2227 \u00ac\u2203 (x : A), x \u2208 B \u2227 R x c \u2227 x \u2260 c\n contradict h3.right with h4\n --h4 : \u00acb = c; Goal : \u2203 (x : A), x \u2208 B \u2227 R x c \u2227 x \u2260 c\n have h5 : R b c := h2.right c h3.left\n show \u2203 (x : A), x \u2208 B \u2227 R x c \u2227 x \u2260 c from\n Exists.intro b (And.intro h2.left (And.intro h5 h4))\n done\n done\n\ntheorem Theorem_4_4_6_3 {A : Type} (R : BinRel A) (B : Set A) (b : A)\n (h1 : total_order R) (h2 : minimalElt R b B) : smallestElt R b B := by\n define at h1 --h1 : partial_order R \u2227 \u2200 (x y : A), R x y \u2228 R y x\n define at h2 --h2 : b \u2208 B \u2227 \u00ac\u2203 (x : A), x \u2208 B \u2227 R x b \u2227 x \u2260 b\n define --Goal : b \u2208 B \u2227 \u2200 (x : A), x \u2208 B \u2192 R b x\n apply And.intro h2.left --Goal : \u2200 (x : A), x \u2208 B \u2192 R b x\n fix x : A\n assume h3 : x \u2208 B --Goal : R b x\n by_cases h4 : x = b\n \u00b7 -- Case 1. h4 : x = b\n rewrite [h4] --Goal : R b b\n have h5 : partial_order R := h1.left\n define at h5\n have h6 : reflexive R := h5.left\n define at h6\n show R b b from h6 b\n done\n \u00b7 -- Case 2. h4 : x \u2260 b\n have h5 : \u2200 (x y : A), R x y \u2228 R y x := h1.right\n have h6 : R x b \u2228 R b x := h5 x b\n have h7 : \u00acR x b := by\n contradict h2.right with h8\n show \u2203 (x : A), x \u2208 B \u2227 R x b \u2227 x \u2260 b from\n Exists.intro x (And.intro h3 (And.intro h8 h4))\n done\n disj_syll h6 h7\n show R b x from h6\n done\n done\n\n/- Section 4.5 -/\n", "theoremStatement": "lemma Lemma_4_5_5_1 {A : Type} (R : BinRel A) (h : equiv_rel R) :\n \u2200 (x : A), x \u2208 equivClass R x ", 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"Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n fix x : A\n define --Goal : R x x\n define at h --h : reflexive R \u2227 symmetric R \u2227 transitive R\n have Rref : reflexive R := h.left\n show R x x from Rref x\n done", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 185}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap7\nnamespace HTPI\n\n/- Definitions -/\ndef fnz (n : Nat) : Int := if 2 \u2223 n then \u2191(n / 2) else -\u2191((n + 1) / 2)\n\ndef fzn (a : Int) : Nat := if a \u2265 0 then 2 * Int.toNat a else 2 * Int.toNat (-a) - 1\n\ndef tri (k : Nat) : Nat := k * (k + 1) / 2\n\ndef fnnn (p : Nat \u00d7 Nat) : Nat := tri (p.1 + p.2) + p.1\n\ndef num_elts_below (A : Set Nat) (m s : Nat) : Prop :=\n match m with\n | 0 => s = 0\n | n + 1 => (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s)\n\ndef enum (A : Set Nat) (s n : Nat) : Prop := n \u2208 A \u2227 num_elts_below A n s\n\ndef unique_val_on_N {U : Type} (R : Rel Nat U) : Prop :=\n \u2200 \u2983n : Nat\u2984 \u2983x1 x2 : U\u2984, R n x1 \u2192 R n x2 \u2192 x1 = x2\n\ndef nat_rel_onto {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203 (n : Nat), R n x\n\ndef fcnl_onto_from_nat {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n unique_val_on_N R \u2227 nat_rel_onto R A\n\ndef fcnl_one_one_to_nat {U : Type} (R : Rel U Nat) (A : Set U) : Prop :=\n fcnl_on R A \u2227 \u2200 \u2983x1 x2 : U\u2984 \u2983n : Nat\u2984,\n (x1 \u2208 A \u2227 R x1 n) \u2192 (x2 \u2208 A \u2227 R x2 n) \u2192 x1 = x2\n\ndef least_rel_to {U : Type} (S : Rel Nat U) (x : U) (n : Nat) : Prop :=\n S n x \u2227 \u2200 (m : Nat), S m x \u2192 n \u2264 m\n\ndef restrict_to {U V : Type} (S : Rel U V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 S x y\n\ndef fqn (q : Rat) : Nat := fnnn (fzn q.num, q.den)\n\ndef Set_rp_below (m : Nat) : Set Nat := {n : Nat | rel_prime m n \u2227 n < m}\n\ndef Set_prod {U V : Type} (A : Set U) (B : Set V) : Set (U \u00d7 V) :=\n {(a, b) : U \u00d7 V | a \u2208 A \u2227 b \u2208 B}\n\nnotation:75 A:75 \" \u00d7\u209b \" B:75 => Set_prod A B\n\ndef Rel_prod {U V W X : Type} (R : Rel U V) (S : Rel W X)\n (p : U \u00d7 W) (q : V \u00d7 X) : Prop := R p.1 q.1 \u2227 S p.2 q.2\n\nnotation:75 R:75 \" \u00d7\u1d63 \" S:75 => Rel_prod R S\n\ndef qr (n a : Nat) : Nat \u00d7 Nat := (a / n, a % n)\n\ndef mod_mod (m n a : Nat) : Nat \u00d7 Nat := (a % m, a % n)\n\ndef enum_union_fam {U : Type}\n (F : Set (Set U)) (f : Set U \u2192 Rel Nat U) (R : Rel Nat (Set U))\n (n : Nat) (a : U) : Prop := \u2203 (p : Nat \u00d7 Nat), fnnn p = n \u2227\n \u2203 A \u2208 F, R p.1 A \u2227 (f A) p.2 a\n\ndef seq {U : Type} (A : Set U) : Set (List U) :=\n {l : List U | \u2200 x \u2208 l, x \u2208 A}\n\ndef seq_by_length {U : Type} (A : Set U) (n : Nat) : Set (List U) :=\n {l : List U | l \u2208 seq A \u2227 l.length = n}\n\ndef seq_cons (U : Type) (p : U \u00d7 (List U)) : List U := p.1 :: p.2\n\ndef sbl_set {U : Type} (A : Set U) : Set (Set (List U)) :=\n {S : Set (List U) | \u2203 (n : Nat), seq_by_length A n = S}\n\ndef rep_common_image\n {U V : Type} (R S : Rel U V) (X0 : Set U) (n : Nat) : Set U :=\n match n with\n | 0 => X0\n | m + 1 => {a : U | \u2203 x \u2208 rep_common_image R S X0 m,\n \u2203 (y : V), R x y \u2227 S a y}\n\ndef cum_rep_image {U V : Type} (R S : Rel U V) (X0 : Set U) : Set U :=\n {a : U | \u2203 (n : Nat), a \u2208 rep_common_image R S X0 n}\n\ndef csb_match {U V : Type} (R S : Rel U V) (X0 : Set U)\n (x : U) (y : V) : Prop := x \u2208 cum_rep_image R S X0 \u2227 R x y \u2228\n x \u2209 cum_rep_image R S X0 \u2227 S x y\n\n/- Section 8.1 -/\n#eval [fnz 0, fnz 1, fnz 2, fnz 3, fnz 4, fnz 5, fnz 6]\n --Answer: [0, -1, 1, -2, 2, -3, 3]\n\n#eval [fzn 0, fzn (-1), fzn 1, fzn (-2), fzn 2, fzn (-3), fzn 3]\n --Answer: [0, 1, 2, 3, 4, 5, 6]\n\nlemma fnz_even (k : Nat) : fnz (2 * k) = \u2191k := by\n have h1 : 2 \u2223 2 * k := by\n apply Exists.intro k\n rfl\n done\n have h2 : fnz (2 * k) = if 2 \u2223 2 * k then \u2191(2 * k / 2)\n else -\u2191((2 * k + 1) / 2) := by rfl\n rewrite [if_pos h1] at h2 --h2 : fnz (2 * k) = \u2191(2 * k / 2)\n have h3 : 0 < 2 := by linarith\n rewrite [Nat.mul_div_cancel_left k h3] at h2\n show fnz (2 * k) = \u2191k from h2\n done\n\nlemma fnz_odd (k : Nat) : fnz (2 * k + 1) = -\u2191(k + 1) := sorry\n\nlemma fzn_nat (k : Nat) : fzn \u2191k = 2 * k := by rfl\n\nlemma fzn_neg_succ_nat (k : Nat) : fzn (-\u2191(k + 1)) = 2 * k + 1 := by rfl\n\n--From exercises of Section 6.1\ntheorem Exercise_6_1_16a1 : \u2200 (n : Nat), nat_even n \u2228 nat_odd n := sorry\n\nlemma fzn_fnz : fzn \u2218 fnz = id := by\n apply funext --Goal : \u2200 (x : Nat), (fzn \u2218 fnz) x = id x\n fix n : Nat\n rewrite [comp_def] --Goal : fzn (fnz n) = id n\n have h1 : nat_even n \u2228 nat_odd n := Exercise_6_1_16a1 n\n by_cases on h1\n \u00b7 -- Case 1. h1 : nat_even n\n obtain (k : Nat) (h2 : n = 2 * k) from h1\n rewrite [h2, fnz_even, fzn_nat]\n rfl\n done\n \u00b7 -- Case 2. h1 : nat_odd n\n obtain (k : Nat) (h2 : n = 2 * k + 1) from h1\n rewrite [h2, fnz_odd, fzn_neg_succ_nat]\n rfl\n done\n done\n\nlemma fnz_fzn : fnz \u2218 fzn = id := sorry\n\nlemma fzn_one_one : one_to_one fzn := Theorem_5_3_3_1 fzn fnz fnz_fzn\n\nlemma fzn_onto : onto fzn := Theorem_5_3_3_2 fzn fnz fzn_fnz\n\nlemma fnz_one_one : one_to_one fnz := Theorem_5_3_3_1 fnz fzn fzn_fnz\n\nlemma fnz_onto : onto fnz := Theorem_5_3_3_2 fnz fzn fnz_fzn\n\nlemma fnnn_def (a b : Nat) : fnnn (a, b) = tri (a + b) + a := by rfl\n\n#eval [fnnn (0, 0), fnnn (0, 1), fnnn (1, 0), fnnn (0, 2), fnnn (1, 1)]\n --Answer: [0, 1, 2, 3, 4]\n\nlemma tri_step (k : Nat) : tri (k + 1) = tri k + k + 1 := sorry\n\nlemma tri_incr {j k : Nat} (h1 : j \u2264 k) : tri j \u2264 tri k := sorry\n\nlemma le_of_fnnn_eq {a1 b1 a2 b2 : Nat}\n (h1 : fnnn (a1, b1) = fnnn (a2, b2)) : a1 + b1 \u2264 a2 + b2 := by\n by_contra h2\n have h3 : a2 + b2 + 1 \u2264 a1 + b1 := by linarith\n have h4 : fnnn (a2, b2) < fnnn (a1, b1) :=\n calc fnnn (a2, b2)\n _ = tri (a2 + b2) + a2 := by rfl\n _ < tri (a2 + b2) + (a2 + b2) + 1 := by linarith\n _ = tri (a2 + b2 + 1) := (tri_step _).symm\n _ \u2264 tri (a1 + b1) := tri_incr h3\n _ \u2264 tri (a1 + b1) + a1 := by linarith\n _ = fnnn (a1, b1) := by rfl\n linarith\n done\n\nlemma fnnn_one_one : one_to_one fnnn := by\n fix (a1, b1) : Nat \u00d7 Nat\n fix (a2, b2) : Nat \u00d7 Nat\n assume h1 : fnnn (a1, b1) = fnnn (a2, b2) --Goal : (a1, b1) = (a2, b2)\n have h2 : a1 + b1 \u2264 a2 + b2 := le_of_fnnn_eq h1\n have h3 : a2 + b2 \u2264 a1 + b1 := le_of_fnnn_eq h1.symm\n have h4 : a1 + b1 = a2 + b2 := by linarith\n rewrite [fnnn_def, fnnn_def, h4] at h1\n --h1 : tri (a2 + b2) + a1 = tri (a2 + b2) + a2\n have h6 : a1 = a2 := Nat.add_left_cancel h1\n rewrite [h6] at h4 --h4 : a2 + b1 = a2 + b2\n have h7 : b1 = b2 := Nat.add_left_cancel h4\n rewrite [h6, h7]\n rfl\n done\n\nlemma fnnn_onto : onto fnnn := by\n define --Goal : \u2200 (y : Nat), \u2203 (x : Nat \u00d7 Nat), fnnn x = y\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro (0, 0)\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2203 (x : Nat \u00d7 Nat), fnnn x = n\n obtain ((a, b) : Nat \u00d7 Nat) (h1 : fnnn (a, b) = n) from ih\n by_cases h2 : b = 0\n \u00b7 -- Case 1. h2 : b = 0\n apply Exists.intro (0, a + 1)\n show fnnn (0, a + 1) = n + 1 from\n calc fnnn (0, a + 1)\n _ = tri (0 + (a + 1)) + 0 := by rfl\n _ = tri (a + 1) := by ring\n _ = tri a + a + 1 := tri_step a\n _ = tri (a + 0) + a + 1 := by ring\n _ = fnnn (a, b) + 1 := by rw [h2, fnnn_def]\n _ = n + 1 := by rw [h1]\n done\n \u00b7 -- Case 2. h2 : b \u2260 0\n obtain (k : Nat) (h3 : b = k + 1) from\n exists_eq_add_one_of_ne_zero h2\n apply Exists.intro (a + 1, k)\n show fnnn (a + 1, k) = n + 1 from\n calc fnnn (a + 1, k)\n _ = tri (a + 1 + k) + (a + 1) := by rfl\n _ = tri (a + (k + 1)) + a + 1 := by ring\n _ = tri (a + b) + a + 1 := by rw [h3]\n _ = fnnn (a, b) + 1 := by rfl\n _ = n + 1 := by rw [h1]\n done\n done\n done\n\nlemma one_one_on_of_one_one {U V : Type} {f : U \u2192 V}\n (h : one_to_one f) (A : Set U) : one_one_on f A := by\n define\n fix x1 : U; fix x2 : U\n assume h1 : x1 \u2208 A\n assume h2 : x2 \u2208 A\n show f x1 = f x2 \u2192 x1 = x2 from h x1 x2\n done\n\nlemma elt_Univ {U : Type} (u : U) :\n u \u2208 Univ U := by trivial\n\ntheorem equinum_Univ {U V : Type} {f : U \u2192 V}\n (h1 : one_to_one f) (h2 : onto f) : Univ U \u223c Univ V := by\n have h3 : image f (Univ U) = Univ V := by\n apply Set.ext\n fix v : V\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h3 : v \u2208 image f (Univ U)\n show v \u2208 Univ V from elt_Univ v\n done\n \u00b7 -- (\u2190)\n assume h3 : v \u2208 Univ V\n obtain (u : U) (h4 : f u = v) from h2 v\n apply Exists.intro u\n apply And.intro _ h4\n show u \u2208 Univ U from elt_Univ u\n done\n done\n show Univ U \u223c Univ V from\n equinum_image (one_one_on_of_one_one h1 (Univ U)) h3\n done\n\ntheorem Z_equinum_N : Univ Int \u223c Univ Nat :=\n equinum_Univ fzn_one_one fzn_onto\n\ntheorem NxN_equinum_N : Univ (Nat \u00d7 Nat) \u223c Univ Nat :=\n equinum_Univ fnnn_one_one fnnn_onto\n\nlemma neb_step (A : Set Nat) (n s : Nat) : num_elts_below A (n + 1) s \u2194\n (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s) := by rfl\n\nlemma neb_step_elt {A : Set Nat} {n : Nat} (h1 : n \u2208 A) (s : Nat) :\n num_elts_below A (n + 1) s \u2194 1 \u2264 s \u2227 num_elts_below A n (s - 1) := by\n rewrite [neb_step]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) \u2228\n \u00acn \u2208 A \u2227 num_elts_below A n s\n by_cases on h2\n \u00b7 -- Case 1. h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n show 1 \u2264 s \u2227 num_elts_below A n (s - 1) from h2.right\n done\n \u00b7 -- Case 2. h2 : \u00acn \u2208 A \u2227 num_elts_below A n s\n show 1 \u2264 s \u2227 num_elts_below A n (s - 1) from absurd h1 h2.left\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n apply Or.inl\n show n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) from And.intro h1 h2\n done\n done\n\nlemma neb_step_not_elt {A : Set Nat} {n : Nat} (h1 : n \u2209 A) (s : Nat) :\n num_elts_below A (n + 1) s \u2194 num_elts_below A n s := by\n rewrite [neb_step]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) \u2228\n \u00acn \u2208 A \u2227 num_elts_below A n s\n by_cases on h2\n \u00b7 -- Case 1. h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n show num_elts_below A n s from absurd h2.left h1\n done\n \u00b7 -- Case 2. h2 : \u00acn \u2208 A \u2227 num_elts_below A n s\n show num_elts_below A n s from h2.right\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : num_elts_below A n s\n apply Or.inr\n show \u00acn \u2208 A \u2227 num_elts_below A n s from And.intro h1 h2\n done\n done\n\nlemma neb_exists (A : Set Nat) :\n \u2200 (n : Nat), \u2203 (s : Nat), num_elts_below A n s := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro 0\n define\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2203 (s : Nat), num_elts_below A n s\n obtain (t : Nat) (h1 : num_elts_below A n t) from ih\n by_cases h2 : n \u2208 A\n \u00b7 -- Case 1. h2 : n \u2208 A\n apply Exists.intro (t + 1)\n rewrite [neb_step_elt h2, Nat.add_sub_cancel]\n apply And.intro _ h1\n linarith\n done\n \u00b7 -- Case 2. h2 : n \u2209 A\n apply Exists.intro t\n rewrite [neb_step_not_elt h2]\n show num_elts_below A n t from h1\n done\n done\n done\n\nlemma neb_unique (A : Set Nat) : \u2200 \u2983n : Nat\u2984, \u2200 \u2983s1 s2 : Nat\u2984,\n num_elts_below A n s1 \u2192 num_elts_below A n s2 \u2192 s1 = s2 := by\n by_induc\n \u00b7 -- Base Case\n fix s1 : Nat; fix s2 : Nat\n assume h1 : num_elts_below A 0 s1\n assume h2 : num_elts_below A 0 s2\n define at h1; define at h2 --h1 : s1 = 0; h2 : s2 = 0\n rewrite [h1, h2]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 \u2983s1 s2 : Nat\u2984,\n num_elts_below A n s1 \u2192 num_elts_below A n s2 \u2192 s1 = s2\n fix s1 : Nat; fix s2 : Nat\n assume h1 : num_elts_below A (n + 1) s1\n assume h2 : num_elts_below A (n + 1) s2\n by_cases h3 : n \u2208 A\n \u00b7 -- Case 1. h3 : n \u2208 A\n rewrite [neb_step_elt h3] at h1\n rewrite [neb_step_elt h3] at h2\n --h1 : 1 \u2264 s1 \u2227 num_elts_below A n (s1 - 1)\n --h2 : 1 \u2264 s2 \u2227 num_elts_below A n (s2 - 1)\n have h4 : s1 - 1 = s2 - 1 := ih h1.right h2.right\n show s1 = s2 from\n calc s1\n _ = s1 - 1 + 1 := (Nat.sub_add_cancel h1.left).symm\n _ = s2 - 1 + 1 := by rw [h4]\n _ = s2 := Nat.sub_add_cancel h2.left\n done\n \u00b7 -- Case 2. h3 : n \u2209 A\n rewrite [neb_step_not_elt h3] at h1 --h1 : num_elts_below A n s1\n rewrite [neb_step_not_elt h3] at h2 --h2 : num_elts_below A n s2\n show s1 = s2 from ih h1 h2\n done\n done\n done\n\nlemma neb_increase {A : Set Nat} {n s : Nat} (h1 : enum A s n) :\n \u2200 \u2983m : Nat\u2984, m \u2265 n + 1 \u2192 \u2200 \u2983t : Nat\u2984, num_elts_below A m t \u2192 s < t := by\n by_induc\n \u00b7 -- Base Case\n define at h1\n fix t : Nat\n assume h2 : num_elts_below A (n + 1) t\n rewrite [neb_step_elt h1.left] at h2\n have h3 : s = t - 1 := neb_unique A h1.right h2.right\n show s < t from\n calc s\n _ = t - 1 := h3\n _ < t - 1 + 1 := by linarith\n _ = t := Nat.sub_add_cancel h2.left\n done\n \u00b7 -- Induction Step\n fix m : Nat\n assume h2 : m \u2265 n + 1\n assume ih : \u2200 \u2983t : Nat\u2984, num_elts_below A m t \u2192 s < t\n fix t : Nat\n assume h3 : num_elts_below A (m + 1) t\n by_cases h4 : m \u2208 A\n \u00b7 -- Case 1. h4 : m \u2208 A\n rewrite [neb_step_elt h4] at h3\n have h5 : s < t - 1 := ih h3.right\n show s < t from\n calc s\n _ < t - 1 := h5\n _ \u2264 t := Nat.sub_le _ _\n done\n \u00b7 -- Case 2. h4 : m \u2209 A\n rewrite [neb_step_not_elt h4] at h3\n show s < t from ih h3\n done\n done\n done\n\nlemma enum_not_skip {A : Set Nat} : \u2200 \u2983m s : Nat\u2984, num_elts_below A m s \u2192\n \u2200 t < s, \u2203 (n : Nat), enum A t n := by\n by_induc\n \u00b7 -- Base Case\n fix s : Nat\n assume h1 : num_elts_below A 0 s\n define at h1\n fix t : Nat\n contrapos\n assume h2 : \u00ac\u2203 (n : Nat), enum A t n\n linarith\n done\n \u00b7 -- Induction Step\n fix m : Nat\n assume ih : \u2200 \u2983s : Nat\u2984, num_elts_below A m s \u2192 \u2200 t < s, \u2203 (n : Nat), enum A t n\n fix s : Nat\n assume h1 : num_elts_below A (m + 1) s\n by_cases h2 : m \u2208 A\n \u00b7 -- Case 1. h2 : m \u2208 A\n rewrite [neb_step_elt h2] at h1\n have h3 : \u2200 t < s - 1, \u2203 (n : Nat), enum A t n := ih h1.right\n fix t : Nat\n assume h4 : t < s\n by_cases h5 : t = s - 1\n \u00b7 -- Case 1.1. h5 : t = s - 1\n apply Exists.intro m\n define\n apply And.intro h2\n rewrite [h5]\n show num_elts_below A m (s - 1) from h1.right\n done\n \u00b7 -- Case 1.2. h5 : t \u2260 s - 1\n have h6 : t \u2264 s - 1 := Nat.le_pred_of_lt h4\n have h7 : t < s - 1 := Nat.lt_of_le_of_ne h6 h5\n show \u2203 (n : Nat), enum A t n from ih h1.right t h7\n done\n done\n \u00b7 -- Case 2. h2 : m \u2209 A\n rewrite [neb_step_not_elt h2] at h1\n show \u2200 t < s, \u2203 (n : Nat), enum A t n from ih h1\n done\n done\n done\n\nlemma enum_le {A : Set Nat} {t n1 n2 : Nat}\n (h1 : enum A t n1) (h2 : enum A t n2) : n1 \u2264 n2 := by\n by_contra h3\n have h4 : n2 + 1 \u2264 n1 := by linarith\n define at h1\n have h5 : t < t := neb_increase h2 h4 h1.right\n linarith\n done\n\nlemma enum_unique (A : Set Nat) (t : Nat) :\n \u2200 \u2983n1 n2 : Nat\u2984, enum A t n1 \u2192 enum A t n2 \u2192 n1 = n2 := by\n fix n1 : Nat; fix n2 : Nat\n assume h1 : enum A t n1\n assume h2 : enum A t n2\n have h3 : n1 \u2264 n2 := enum_le h1 h2\n have h4 : n2 \u2264 n1 := enum_le h2 h1\n linarith\n done\n\nlemma inv_enum_fcnl (A : Set Nat) : fcnl_on (invRel (enum A)) A := by\n define\n fix n : Nat\n assume h1 : n \u2208 A\n exists_unique\n \u00b7 -- Existence\n obtain (s : Nat) (h2 : num_elts_below A n s) from neb_exists A n\n apply Exists.intro s\n define\n show n \u2208 A \u2227 num_elts_below A n s from And.intro h1 h2\n done\n \u00b7 -- Uniqueness\n fix s1 : Nat; fix s2 : Nat\n assume h2 : invRel (enum A) n s1\n assume h3 : invRel (enum A) n s2\n define at h2; define at h3\n show s1 = s2 from neb_unique A h2.right h3.right\n done\n done\n\nlemma bdd_subset_nat_match {A : Set Nat} {m s : Nat}\n (h1 : \u2200 n \u2208 A, n < m) (h2 : num_elts_below A m s) :\n matching (enum A) (I s) A := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix t : Nat; fix n : Nat\n assume h3 : enum A t n\n have h4 : \u2200 \u2983m : Nat\u2984, m \u2265 n + 1 \u2192 \u2200 \u2983t_1 : Nat\u2984, num_elts_below A m t_1 \u2192 t < t_1 :=\n neb_increase h3\n define at h3\n apply And.intro _ h3.left\n define\n have h5 : n < m := h1 n h3.left\n have h6 : m \u2265 n + 1 := h5\n show t < s from h4 h6 h2\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- proof of fcnl_on (enum A) (I s)\n define\n fix t : Nat\n assume h3 : t \u2208 I s\n define at h3\n exists_unique\n \u00b7 -- Existence\n show \u2203 (y : Nat), enum A t y from enum_not_skip h2 t h3\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A t y_1 \u2192 enum A t y_2 \u2192 y_1 = y_2 from\n enum_unique A t\n done\n done\n \u00b7 -- Proof of fcnl_on (invRel (enum A)) A\n show fcnl_on (invRel (enum A)) A from inv_enum_fcnl A\n done\n done\n done\n\nlemma bdd_subset_nat {A : Set Nat} {m s : Nat}\n (h1 : \u2200 n \u2208 A, n < m) (h2 : num_elts_below A m s) :\n I s \u223c A := Exists.intro (enum A) (bdd_subset_nat_match h1 h2)\n\nlemma enum_fcnl_of_unbdd {A : Set Nat} (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n fcnl_on (enum A) (Univ Nat) := by\n define\n by_induc\n \u00b7 -- Base Case\n assume h2 : 0 \u2208 Univ Nat\n exists_unique\n \u00b7 -- Existence\n obtain (n : Nat) (h3 : n \u2208 A \u2227 n \u2265 0) from h1 0\n obtain (s : Nat) (h4 : num_elts_below A (n + 1) s) from neb_exists A (n + 1)\n have h5 : \u2200 (t : Nat), t < s \u2192 \u2203 (n : Nat), enum A t n := enum_not_skip h4\n rewrite [neb_step_elt h3.left] at h4\n show \u2203 (y : Nat), enum A 0 y from h5 0 h4.left\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A 0 y_1 \u2192 enum A 0 y_2 \u2192 y_1 = y_2 from enum_unique A 0\n done\n done\n \u00b7 -- Induction Step\n fix s : Nat\n assume ih : s \u2208 Univ Nat \u2192 \u2203! (y : Nat), enum A s y\n assume h2 : s + 1 \u2208 Univ Nat\n exists_unique\n \u00b7 -- Existence\n have h3 : s \u2208 Univ Nat := elt_Univ s\n obtain (m : Nat) (h4 : enum A s m)\n (h5 : \u2200 (y_1 y_2 : Nat), enum A s y_1 \u2192 enum A s y_2 \u2192 y_1 = y_2) from ih h3\n obtain (n : Nat) (h6 : n \u2208 A \u2227 n \u2265 m + 1) from h1 (m + 1)\n obtain (t : Nat) (h7 : num_elts_below A n t) from neb_exists A n\n have h8 : s < t := neb_increase h4 h6.right h7\n have h9 : s + 1 < t \u2228 s + 1 = t := Nat.lt_or_eq_of_le h8\n by_cases on h9\n \u00b7 -- Case 1. h9 : s + 1 < t\n show \u2203 (y : Nat), enum A (s + 1) y from enum_not_skip h7 (s + 1) h9\n done\n \u00b7 -- Case 2. h9 : s + 1 = t\n rewrite [h9]\n apply Exists.intro n\n define\n show n \u2208 A \u2227 num_elts_below A n t from And.intro h6.left h7\n done\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A (s + 1) y_1 \u2192 enum A (s + 1) y_2 \u2192 y_1 = y_2\n from enum_unique A (s + 1)\n done\n done\n done\n\nlemma unbdd_subset_nat_match {A : Set Nat}\n (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n matching (enum A) (Univ Nat) A := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix s : Nat; fix n : Nat\n assume h2 : enum A s n\n define at h2\n apply And.intro (elt_Univ s) h2.left\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on (enum A)\n show fcnl_on (enum A) (Univ Nat) from enum_fcnl_of_unbdd h1\n done\n \u00b7 -- Proof of fcnl_on (invRel (enum A))\n show fcnl_on (invRel (enum A)) A from inv_enum_fcnl A\n done\n done\n done\n\nlemma unbdd_subset_nat {A : Set Nat}\n (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n denum A := Exists.intro (enum A) (unbdd_subset_nat_match h1)\n\nlemma subset_nat_ctble (A : Set Nat) : ctble A := by\n define --Goal : finite A \u2228 denum A\n by_cases h1 : \u2203 (m : Nat), \u2200 n \u2208 A, n < m\n \u00b7 -- Case 1. h1 : \u2203 (m : Nat), \u2200 n \u2208 A, n < m\n apply Or.inl --Goal : finite A\n obtain (m : Nat) (h2 : \u2200 n \u2208 A, n < m) from h1\n obtain (s : Nat) (h3 : num_elts_below A m s) from neb_exists A m\n apply Exists.intro s\n show I s \u223c A from bdd_subset_nat h2 h3\n done\n \u00b7 -- Case 2. h1 : \u00ac\u2203 (m : Nat), \u2200 n \u2208 A, n < m\n apply Or.inr --Goal : denum A\n push_neg at h1\n --This tactic converts h1 to \u2200 (m : Nat), \u2203 n \u2208 A, m \u2264 n\n show denum A from unbdd_subset_nat h1\n done\n done\n\nlemma ctble_of_equinum_ctble {U V : Type} {A : Set U} {B : Set V}\n (h1 : A \u223c B) (h2 : ctble A) : ctble B := sorry\n\nlemma ctble_iff_equinum_set_nat {U : Type} (A : Set U) :\n ctble A \u2194 \u2203 (I : Set Nat), I \u223c A := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : ctble A\n define at h1 --h1 : finite A \u2228 denum A\n by_cases on h1\n \u00b7 -- Case 1. h1 : finite A\n define at h1 --h1 : \u2203 (n : Nat), I n \u223c A\n obtain (n : Nat) (h2 : I n \u223c A) from h1\n show \u2203 (I : Set Nat), I \u223c A from Exists.intro (I n) h2\n done\n \u00b7 -- Case 2. h1 : denum A\n rewrite [denum_def] at h1 --h1 : Univ Nat \u223c A\n show \u2203 (I : Set Nat), I \u223c A from Exists.intro (Univ Nat) h1\n done\n done\n \u00b7 -- (\u2190)\n assume h1 : \u2203 (I : Set Nat), I \u223c A\n obtain (I : Set Nat) (h2 : I \u223c A) from h1\n have h3 : ctble I := subset_nat_ctble I\n show ctble A from ctble_of_equinum_ctble h2 h3\n done\n done\n\ntheorem Theorem_8_1_5_1_to_2 {U : Type} {A : Set U} (h1 : ctble A) :\n \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A := by\n rewrite [ctble_iff_equinum_set_nat] at h1\n obtain (I : Set Nat) (h2 : I \u223c A) from h1\n obtain (R : Rel Nat U) (h3 : matching R I A) from h2\n define at h3\n --h3 : rel_within R I A \u2227 fcnl_on R I \u2227 fcnl_on (invRel R) A\n apply Exists.intro R\n define --Goal : unique_val_on_N R \u2227 nat_rel_onto R A\n apply And.intro\n \u00b7 -- Proof of unique_val_on_N R\n define\n fix n : Nat; fix x1 : U; fix x2 : U\n assume h4 : R n x1\n assume h5 : R n x2 --Goal : x1 = x2\n have h6 : n \u2208 I \u2227 x1 \u2208 A := h3.left h4\n show x1 = x2 from fcnl_unique h3.right.left h6.left h4 h5\n done\n \u00b7 -- Proof of nat_rel_onto R A\n define\n fix x : U\n assume h4 : x \u2208 A --Goal : \u2203 (n : Nat), R n x\n show \u2203 (n : Nat), R n x from fcnl_exists h3.right.right h4\n done\n done\n\nlemma exists_least_rel_to {U : Type} {S : Rel Nat U} {x : U}\n (h1 : \u2203 (n : Nat), S n x) : \u2203 (n : Nat), least_rel_to S x n := by\n set W : Set Nat := {n : Nat | S n x}\n have h2 : \u2203 (n : Nat), n \u2208 W := h1\n show \u2203 (n : Nat), least_rel_to S x n from well_ord_princ W h2\n done\n\ntheorem Theorem_8_1_5_2_to_3 {U : Type} {A : Set U}\n (h1 : \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A) :\n \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A := by\n obtain (S : Rel Nat U) (h2 : fcnl_onto_from_nat S A) from h1\n define at h2 --h2 : unique_val_on_N S \u2227 nat_rel_onto S A\n set R : Rel U Nat := least_rel_to S\n apply Exists.intro R\n define\n apply And.intro\n \u00b7 -- Proof of fcnl_on R A\n define\n fix x : U\n assume h4 : x \u2208 A --Goal : \u2203! (y : Nat), R x y\n exists_unique\n \u00b7 -- Existence\n have h5 : \u2203 (n : Nat), S n x := h2.right h4\n show \u2203 (n : Nat), R x n from exists_least_rel_to h5\n done\n \u00b7 -- Uniqueness\n fix n1 : Nat; fix n2 : Nat\n assume h5 : R x n1\n assume h6 : R x n2 --Goal : n1 = n2\n define at h5 --h5 : S n1 x \u2227 \u2200 (m : Nat), S m x \u2192 n1 \u2264 m\n define at h6 --h6 : S n2 x \u2227 \u2200 (m : Nat), S m x \u2192 n2 \u2264 m\n have h7 : n1 \u2264 n2 := h5.right n2 h6.left\n have h8 : n2 \u2264 n1 := h6.right n1 h5.left\n linarith\n done\n done\n \u00b7 -- Proof of one-to-one\n fix x1 : U; fix x2 : U; fix n : Nat\n assume h4 : x1 \u2208 A \u2227 R x1 n\n assume h5 : x2 \u2208 A \u2227 R x2 n\n have h6 : R x1 n := h4.right\n have h7 : R x2 n := h5.right\n define at h6 --h6 : S n x1 \u2227 \u2200 (m : Nat), S m x1 \u2192 n \u2264 m\n define at h7 --h7 : S n x2 \u2227 \u2200 (m : Nat), S m x2 \u2192 n \u2264 m\n show x1 = x2 from h2.left h6.left h7.left\n done\n done\n\ntheorem Theorem_8_1_5_3_to_1 {U : Type} {A : Set U}\n (h1 : \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A) :\n ctble A := by\n obtain (S : Rel U Nat) (h2 : fcnl_one_one_to_nat S A) from h1\n define at h2 --h2 : fcnl_on S A \u2227 \u2200 \u2983x1 x2 : U\u2984 \u2983n : Nat\u2984,\n --x1 \u2208 A \u2227 S x1 n \u2192 x2 \u2208 A \u2227 S x2 n \u2192 x1 = x2\n rewrite [ctble_iff_equinum_set_nat] --Goal : \u2203 (I : Set Nat), I \u223c A\n set R : Rel Nat U := invRel (restrict_to S A)\n set I : Set Nat := {n : Nat | \u2203 (x : U), R n x}\n apply Exists.intro I\n define --Goal : \u2203 (R : Rel Nat U), matching R I A\n apply Exists.intro R\n define\n apply And.intro\n \u00b7 -- Proof of rel_within R I A\n define\n fix n : Nat; fix x : U\n assume h3 : R n x --Goal : n \u2208 I \u2227 x \u2208 A\n apply And.intro\n \u00b7 -- Proof that n \u2208 I\n define --Goal : \u2203 (x : U), R n x\n show \u2203 (x : U), R n x from Exists.intro x h3\n done\n \u00b7 -- Proof that x \u2208 A\n define at h3 --h3 : x \u2208 A \u2227 S x n\n show x \u2208 A from h3.left\n done\n done\n \u00b7 -- Proofs of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on R I\n define\n fix n : Nat\n assume h3 : n \u2208 I --Goal : \u2203! (y : U), R n y\n exists_unique\n \u00b7 -- Existence\n define at h3 --h3 : \u2203 (x : U), R n x\n show \u2203 (y : U), R n y from h3\n done\n \u00b7 -- Uniqueness\n fix x1 : U; fix x2 : U\n assume h4 : R n x1\n assume h5 : R n x2\n define at h4 --h4 : x1 \u2208 A \u2227 S x1 n;\n define at h5 --h5 : x2 \u2208 A \u2227 S x2 n\n show x1 = x2 from h2.right h4 h5\n done\n done\n \u00b7 -- Proof of fcnl_on (invRel R) A\n define\n fix x : U\n assume h3 : x \u2208 A --Goal : \u2203! (y : Nat), invRel R x y\n exists_unique\n \u00b7 -- Existence\n obtain (y : Nat) (h4 : S x y) from fcnl_exists h2.left h3\n apply Exists.intro y\n define\n show x \u2208 A \u2227 S x y from And.intro h3 h4\n done\n \u00b7 -- Uniqueness\n fix n1 : Nat; fix n2 : Nat\n assume h4 : invRel R x n1\n assume h5 : invRel R x n2 --Goal : n1 = n2\n define at h4 --h4 : x \u2208 A \u2227 S x n1\n define at h5 --h5 : x \u2208 A \u2227 S x n2\n show n1 = n2 from fcnl_unique h2.left h3 h4.right h5.right\n done\n done\n done\n done\n\ntheorem Theorem_8_1_5_2 {U : Type} (A : Set U) :\n ctble A \u2194 \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : ctble A\n show \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A from\n Theorem_8_1_5_1_to_2 h1\n done\n \u00b7 -- (\u2190)\n assume h1 : \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A\n have h2 : \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A :=\n Theorem_8_1_5_2_to_3 h1\n show ctble A from Theorem_8_1_5_3_to_1 h2\n done\n done\n\ntheorem Theorem_8_1_5_3 {U : Type} (A : Set U) :\n ctble A \u2194 \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : ctble A\n have h2 : \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A :=\n Theorem_8_1_5_1_to_2 h1\n show \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A from\n Theorem_8_1_5_2_to_3 h2\n done\n \u00b7 -- (\u2190)\n assume h1 : \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A\n show ctble A from Theorem_8_1_5_3_to_1 h1\n done\n done\n\nlemma fqn_def (q : Rat) : fqn q = fnnn (fzn q.num, q.den) := by rfl\n\nlemma fqn_one_one : one_to_one fqn := by\n define\n fix q1 : Rat; fix q2 : Rat\n assume h1 : fqn q1 = fqn q2\n rewrite [fqn_def, fqn_def] at h1\n --h1 : fnnn (fzn q1.num, q1.den) = fnnn (fzn q2.num, q2.den)\n have h2 : (fzn q1.num, q1.den) = (fzn q2.num, q2.den) :=\n fnnn_one_one _ _ h1\n have h3 : fzn q1.num = fzn q2.num \u2227 q1.den = q2.den :=\n Prod.mk.inj h2\n have h4 : q1.num = q2.num := fzn_one_one _ _ h3.left\n show q1 = q2 from Rat.ext h4 h3.right\n done\n\nlemma image_fqn_unbdd :\n \u2200 (m : Nat), \u2203 n \u2208 image fqn (Univ Rat), n \u2265 m := by\n fix m : Nat\n set n : Nat := fqn \u2191m\n apply Exists.intro n\n apply And.intro\n \u00b7 -- Proof that n \u2208 image fqn (Univ Rat)\n define\n apply Exists.intro \u2191m\n apply And.intro (elt_Univ (\u2191m : Rat))\n rfl\n done\n \u00b7 -- Proof that n \u2265 m\n show n \u2265 m from\n calc n\n _ = tri (2 * m + 1) + 2 * m := by rfl\n _ \u2265 m := by linarith\n done\n done\n\ntheorem Theorem_8_1_6 : denum (Univ Rat) := by\n set I : Set Nat := image fqn (Univ Rat)\n have h1 : Univ Nat \u223c I := unbdd_subset_nat image_fqn_unbdd\n have h2 : image fqn (Univ Rat) = I := by rfl\n have h3 : Univ Rat \u223c I :=\n equinum_image (one_one_on_of_one_one fqn_one_one (Univ Rat)) h2\n have h4 : I \u223c Univ Rat := Theorem_8_1_3_2 h3\n show denum (Univ Rat) from Theorem_8_1_3_3 h1 h4\n done\n\n/- Section 8.1\u00bd -/\nlemma eq_zero_of_I_zero_equinum {n : Nat} (h1 : I 0 \u223c I n) : n = 0 := by\n rewrite [\u2190numElts_def, zero_elts_iff_empty] at h1\n --h1 : empty (I n)\n contradict h1 with h2 --Goal : \u2203 (x : Nat), x \u2208 I n\n apply Exists.intro 0\n define\n show 0 < n from Nat.pos_of_ne_zero h2\n done\n\ntheorem eq_of_I_equinum : \u2200 \u2983m n : Nat\u2984, I m \u223c I n \u2192 m = n := by\n by_induc\n \u00b7 -- Base Case\n fix n : Nat\n assume h1 : I 0 \u223c I n\n show 0 = n from (eq_zero_of_I_zero_equinum h1).symm\n done\n \u00b7 -- Induction Step\n fix m : Nat\n assume ih : \u2200 \u2983n : Nat\u2984, I m \u223c I n \u2192 m = n\n fix n : Nat\n assume h1 : I (m + 1) \u223c I n --Goal : m + 1 = n\n have h2 : n \u2260 0 := by\n by_contra h2\n have h3 : I n \u223c I (m + 1) := Theorem_8_1_3_2 h1\n rewrite [h2] at h3\n have h4 : m + 1 = 0 := eq_zero_of_I_zero_equinum h3\n linarith\n done\n obtain (k : Nat) (h3 : n = k + 1) from exists_eq_add_one_of_ne_zero h2\n rewrite [h3] at h1 --h1 : I (m + 1) \u223c I (k + 1)\n rewrite [h3] --Goal : m + 1 = k + 1\n have h4 : m \u2208 I (m + 1) := I_max m\n have h5 : k \u2208 I (k + 1) := I_max k\n have h6 : I (m + 1) \\ {m} \u223c I (k + 1) \\ {k} :=\n remove_one_equinum h1 h4 h5\n rewrite [I_diff, I_diff] at h6 --h6 : I m \u223c I k\n have h7 : m = k := ih h6\n rewrite [h7]\n rfl\n done\n done\n\ntheorem numElts_unique {U : Type} {A : Set U} {m n : Nat}\n (h1 : numElts A m) (h2 : numElts A n) : m = n := by\n rewrite [numElts_def] at h1 --h1 : I m \u223c A\n rewrite [numElts_def] at h2 --h2 : I n \u223c A\n have h3 : A \u223c I n := Theorem_8_1_3_2 h2\n have h4 : I m \u223c I n := Theorem_8_1_3_3 h1 h3\n show m = n from eq_of_I_equinum h4\n done\n\nlemma Set_rp_below_def (a m : Nat) :\n a \u2208 Set_rp_below m \u2194 rel_prime m a \u2227 a < m := by rfl\n\nlemma neb_nrpb (m : Nat) : \u2200 \u2983k : Nat\u2984, k \u2264 m \u2192\n num_elts_below (Set_rp_below m) k (num_rp_below m k) := sorry\n\nlemma neb_phi (m : Nat) :\n num_elts_below (Set_rp_below m) m (phi m) := by\n rewrite [phi_def]\n have h1 : m \u2264 m := by linarith\n show num_elts_below (Set_rp_below m) m (num_rp_below m m) from\n neb_nrpb m h1\n done\n\nlemma phi_is_numElts (m : Nat) :\n numElts (Set_rp_below m) (phi m) := by\n rewrite [numElts_def] --Goal : I (phi m) \u223c Set_rp_below m\n have h1 : \u2200 n \u2208 Set_rp_below m, n < m := by\n fix n : Nat\n assume h2 : n \u2208 Set_rp_below m\n define at h2\n show n < m from h2.right\n done\n have h2 : num_elts_below (Set_rp_below m) m (phi m) := neb_phi m\n show I (phi m) \u223c Set_rp_below m from bdd_subset_nat h1 h2\n done\n\nlemma Lemma_7_4_7_aux {m n : Nat} {s t : Int}\n (h : s * m + t * n = 1) (a b : Nat) :\n t * n * a + s * m * b \u2261 a (MOD m) := by\n define\n apply Exists.intro (s * (b - a))\n show t * n * a + s * m * b - a = m * (s * (b - a)) from\n calc t * n * a + s * m * b - a\n _ = (t * n - 1) * a + s * m * b := by ring\n _ = (t * n - (s * m + t * n)) * a + s * m * b := by rw [h]\n _ = m * (s * (b - a)) := by ring\n done\n\nlemma Lemma_7_4_7 {m n : Nat} [NeZero m] [NeZero n]\n (h1 : rel_prime m n) (a b : Nat) :\n \u2203 (r : Nat), r < m * n \u2227 r \u2261 a (MOD m) \u2227 r \u2261 b (MOD n) := by\n set s : Int := gcd_c1 m n\n set t : Int := gcd_c2 m n\n have h4 : s * m + t * n = gcd m n := gcd_lin_comb n m\n define at h1 --h1 : gcd m n = 1\n rewrite [h1, Nat.cast_one] at h4 --h4 : s * m + t * n = 1\n set x : Int := t * n * a + s * m * b\n have h5 : x \u2261 a (MOD m) := Lemma_7_4_7_aux h4 a b\n rewrite [add_comm] at h4 --h4 : t * n + s * m = 1\n have h6 : s * m * b + t * n * a \u2261 b (MOD n) :=\n Lemma_7_4_7_aux h4 b a\n have h7 : s * m * b + t * n * a = x := by ring\n rewrite [h7] at h6 --h6 : x \u2261 b (MOD n)\n have h8 : m * n \u2260 0 := mul_ne_zero (NeZero.ne m) (NeZero.ne n)\n rewrite [\u2190neZero_iff] at h8 --h8 : NeZero (m * n)\n have h9 : 0 \u2264 x % \u2191(m * n) \u2227 x % \u2191(m * n) < \u2191(m * n) \u2227\n x \u2261 x % \u2191(m * n) (MOD m * n) := mod_cmpl_res (m * n) x\n have h10 : x % \u2191(m * n) < \u2191(m * n) \u2227\n x \u2261 x % \u2191(m * n) (MOD m * n) := h9.right\n set r : Nat := Int.toNat (x % \u2191(m * n))\n have h11 : x % \u2191(m * n) = \u2191r := (Int.toNat_of_nonneg h9.left).symm\n rewrite [h11, Nat.cast_lt] at h10 --h10 : r < m * n \u2227 x \u2261 r (MOD m * n)\n apply Exists.intro r\n apply And.intro h10.left\n have h12 : r \u2261 x (MOD (m * n)) := congr_symm h10.right\n rewrite [Lemma_7_4_5 _ _ h1] at h12 --h12 : r \u2261 x (MOD m) \u2227 r \u2261 x (MOD n)\n apply And.intro\n \u00b7 -- Proof that r \u2261 a (MOD m)\n show r \u2261 a (MOD m) from congr_trans h12.left h5\n done\n \u00b7 -- Proof that r \u2261 b (MOD n)\n show r \u2261 b (MOD n) from congr_trans h12.right h6\n done\n done\n\nlemma Set_prod_def {U V : Type} (A : Set U) (B : Set V) (a : U) (b : V) :\n (a, b) \u2208 A \u00d7\u209b B \u2194 a \u2208 A \u2227 b \u2208 B := by rfl\n\nlemma Rel_prod_def {U V W X : Type} (R : Rel U V) (S : Rel W X)\n (u : U) (v : V) (w : W) (x : X) :\n (R \u00d7\u1d63 S) (u, w) (v, x) \u2194 R u v \u2227 S w x := by rfl\n\nlemma prod_match {U V W X : Type}\n {A : Set U} {B : Set V} {C : Set W} {D : Set X}\n {R : Rel U V} {S : Rel W X}\n (h1 : matching R A B) (h2 : matching S C D) :\n matching (R \u00d7\u1d63 S) (A \u00d7\u209b C) (B \u00d7\u209b D) := sorry\n\ntheorem Theorem_8_1_2_1\n {U V W X : Type} {A : Set U} {B : Set V} {C : Set W} {D : Set X}\n (h1 : A \u223c B) (h2 : C \u223c D) : A \u00d7\u209b C \u223c B \u00d7\u209b D := by\n obtain (R : Rel U V) (h3 : matching R A B) from h1\n obtain (S : Rel W X) (h4 : matching S C D) from h2\n apply Exists.intro (R \u00d7\u1d63 S)\n show matching (R \u00d7\u1d63 S) (A \u00d7\u209b C) (B \u00d7\u209b D) from prod_match h3 h4\n done\n\nlemma qr_def (n a : Nat) : qr n a = (a / n, a % n) := by rfl\n\nlemma qr_one_one (n : Nat) : one_to_one (qr n) := by\n define\n fix a1 : Nat; fix a2 : Nat\n assume h1 : qr n a1 = qr n a2 --Goal : a1 = a2\n rewrite [qr_def, qr_def] at h1\n have h2 : a1 / n = a2 / n \u2227 a1 % n = a2 % n := Prod.mk.inj h1\n show a1 = a2 from\n calc a1\n _ = n * (a1 / n) + a1 % n := (Nat.div_add_mod a1 n).symm\n _ = n * (a2 / n) + a2 % n := by rw [h2.left, h2.right]\n _ = a2 := Nat.div_add_mod a2 n\n done\n\nlemma qr_image (m n : Nat) :\n image (qr n) (I (m * n)) = I m \u00d7\u209b I n := sorry\n\nlemma I_prod (m n : Nat) : I (m * n) \u223c I m \u00d7\u209b I n := equinum_image\n (one_one_on_of_one_one (qr_one_one n) (I (m * n))) (qr_image m n)\n\ntheorem numElts_prod {U V : Type} {A : Set U} {B : Set V} {m n : Nat}\n (h1 : numElts A m) (h2 : numElts B n) : numElts (A \u00d7\u209b B) (m * n) := by\n rewrite [numElts_def] at h1 --h1 : I m \u223c A\n rewrite [numElts_def] at h2 --h2 : I n \u223c B\n rewrite [numElts_def] --Goal : I (m * n) \u223c A \u00d7\u209b B\n have h3 : I m \u00d7\u209b I n \u223c A \u00d7\u209b B := Theorem_8_1_2_1 h1 h2\n have h4 : I (m * n) \u223c I m \u00d7\u209b I n := I_prod m n\n show I (m * n) \u223c A \u00d7\u209b B from Theorem_8_1_3_3 h4 h3\n done\n\nlemma mod_mod_def (m n a : Nat) : mod_mod m n a = (a % m, a % n) := by rfl\n\n--From exercises of Section 7.3\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n/- Stated in Chap7.lean\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n-/\n\n--From exercises of Section 7.4\nlemma Lemma_7_4_6 {a b c : Nat} :\n rel_prime (a * b) c \u2194 rel_prime a c \u2227 rel_prime b c := sorry\n\nlemma left_NeZero_of_mul {m n : Nat} (h : m * n \u2260 0) : NeZero m :=\n neZero_iff.rtl (left_ne_zero_of_mul h)\n\nlemma right_NeZero_of_mul {m n : Nat} (h : m * n \u2260 0) : NeZero n :=\n neZero_iff.rtl (right_ne_zero_of_mul h)\n\nlemma mod_mod_one_one_on {m n : Nat} (h1 : rel_prime m n) :\n one_one_on (mod_mod m n) (Set_rp_below (m * n)) := by\n define\n fix a1 : Nat; fix a2 : Nat\n assume h2 : a1 \u2208 Set_rp_below (m * n)\n assume h3 : a2 \u2208 Set_rp_below (m * n)\n assume h4 : mod_mod m n a1 = mod_mod m n a2 --Goal : a1 = a2\n define at h2; define at h3\n rewrite [mod_mod_def, mod_mod_def] at h4\n have h5 : a1 % m = a2 % m \u2227 a1 % n = a2 % n := Prod.mk.inj h4\n have h6 : m * n \u2260 0 := by linarith\n have h7 : NeZero m := left_NeZero_of_mul h6\n have h8 : NeZero n := right_NeZero_of_mul h6\n rewrite [\u2190congr_iff_mod_eq_Nat, \u2190congr_iff_mod_eq_Nat] at h5\n --h5 : \u2191a1 \u2261 \u2191a2 (MOD m) \u2227 \u2191a1 \u2261 \u2191a2 (MOD n)\n rewrite [\u2190Lemma_7_4_5 _ _ h1] at h5 --h5 : \u2191a1 \u2261 \u2191a2 (MOD m * n)\n rewrite [congr_iff_mod_eq_Nat] at h5 --h5 : a1 % (m * n) = a2 % (m * n)\n rewrite [Nat.mod_eq_of_lt h2.right, Nat.mod_eq_of_lt h3.right] at h5\n show a1 = a2 from h5\n done\n\nlemma mod_elt_Set_rp_below {a m : Nat} [NeZero m] (h1 : rel_prime m a) :\n a % m \u2208 Set_rp_below m := by\n define --Goal : rel_prime m (a % m) \u2227 a % m < m\n rewrite [rel_prime_mod] --Goal : rel_prime m a \u2227 a % m < m\n show rel_prime m a \u2227 a % m < m from\n And.intro h1 (mod_nonzero_lt a (NeZero.ne m))\n done\n\nlemma mod_mod_image {m n : Nat} (h1 : rel_prime m n) :\n image (mod_mod m n) (Set_rp_below (m * n)) =\n (Set_rp_below m) \u00d7\u209b (Set_rp_below n) := by\n apply Set.ext\n fix (b, c) : Nat \u00d7 Nat\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : (b, c) \u2208 image (mod_mod m n) (Set_rp_below (m * n))\n define at h2\n obtain (a : Nat)\n (h3 : a \u2208 Set_rp_below (m * n) \u2227 mod_mod m n a = (b, c)) from h2\n rewrite [Set_rp_below_def, mod_mod_def] at h3\n have h4 : rel_prime (m * n) a := h3.left.left\n rewrite [Lemma_7_4_6] at h4 --h4 : rel_prime m a \u2227 rel_prime n a\n have h5 : a % m = b \u2227 a % n = c := Prod.mk.inj h3.right\n define\n rewrite [\u2190h5.left, \u2190h5.right]\n --Goal : a % m \u2208 Set_rp_below m \u2227 a % n \u2208 Set_rp_below n\n have h6 : m * n \u2260 0 := by linarith\n have h7 : NeZero m := left_NeZero_of_mul h6\n have h8 : NeZero n := right_NeZero_of_mul h6\n apply And.intro\n \u00b7 -- Proof that a % m \u2208 Set_rp_below m\n show a % m \u2208 Set_rp_below m from mod_elt_Set_rp_below h4.left\n done\n \u00b7 -- Proof that a % n \u2208 Set_rp_below n\n show a % n \u2208 Set_rp_below n from mod_elt_Set_rp_below h4.right\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : (b, c) \u2208 Set_rp_below m \u00d7\u209b Set_rp_below n\n rewrite [Set_prod_def, Set_rp_below_def, Set_rp_below_def] at h2\n --h2 : (rel_prime m b \u2227 b < m) \u2227 (rel_prime n c \u2227 c < n)\n define\n have h3 : m \u2260 0 := by linarith\n have h4 : n \u2260 0 := by linarith\n rewrite [\u2190neZero_iff] at h3\n rewrite [\u2190neZero_iff] at h4\n obtain (a : Nat) (h5 : a < m * n \u2227 a \u2261 b (MOD m) \u2227 a \u2261 c (MOD n))\n from Lemma_7_4_7 h1 b c\n apply Exists.intro a\n apply And.intro\n \u00b7 -- Proof of a \u2208 Set_rp_below (m * n)\n define --Goal : rel_prime (m * n) a \u2227 a < m * n\n apply And.intro _ h5.left\n rewrite [Lemma_7_4_6] --Goal : rel_prime m a \u2227 rel_prime n a\n rewrite [congr_rel_prime h5.right.left,\n congr_rel_prime h5.right.right]\n show rel_prime m b \u2227 rel_prime n c from\n And.intro h2.left.left h2.right.left\n done\n \u00b7 -- Proof of mod_mod m n a = (b, c)\n rewrite [congr_iff_mod_eq_Nat, congr_iff_mod_eq_Nat] at h5\n rewrite [mod_mod_def, h5.right.left, h5.right.right]\n --Goal : (b % m, c % n) = (b, c)\n rewrite [Nat.mod_eq_of_lt h2.left.right,\n Nat.mod_eq_of_lt h2.right.right]\n rfl\n done\n done\n done\n\nlemma Set_rp_below_prod {m n : Nat} (h1 : rel_prime m n) :\n Set_rp_below (m * n) \u223c (Set_rp_below m) \u00d7\u209b (Set_rp_below n) :=\n equinum_image (mod_mod_one_one_on h1) (mod_mod_image h1)\n\nlemma eq_numElts_of_equinum {U V : Type} {A : Set U} {B : Set V} {n : Nat}\n (h1 : A \u223c B) (h2 : numElts A n) : numElts B n := by\n rewrite [numElts_def] at h2 --h2 : I n \u223c A\n rewrite [numElts_def] --Goal : I n \u223c B\n show I n \u223c B from Theorem_8_1_3_3 h2 h1\n done\n\ntheorem Theorem_7_4_4 {m n : Nat} (h1 : rel_prime m n) :\n phi (m * n) = (phi m) * (phi n) := by\n have h2 : numElts (Set_rp_below m) (phi m) := phi_is_numElts m\n have h3 : numElts (Set_rp_below n) (phi n) := phi_is_numElts n\n have h4 : numElts (Set_rp_below (m * n)) (phi (m * n)) :=\n phi_is_numElts (m * n)\n have h5 : numElts (Set_rp_below m \u00d7\u209b Set_rp_below n) (phi (m * n)) :=\n eq_numElts_of_equinum (Set_rp_below_prod h1) h4\n have h6 : numElts (Set_rp_below m \u00d7\u209b Set_rp_below n) (phi m * phi n) :=\n numElts_prod h2 h3\n show phi (m * n) = phi m * phi n from numElts_unique h5 h6\n done\n\n/- Section 8.2 -/\n--From exercises of Section 8.1\ntheorem Exercise_8_1_17 {U : Type} {A B : Set U}\n (h1 : B \u2286 A) (h2 : ctble A) : ctble B := sorry\n\ntheorem Theorem_8_2_1_1 {U V : Type} {A : Set U} {B : Set V}\n (h1 : ctble A) (h2 : ctble B) : ctble (A \u00d7\u209b B) := by\n rewrite [ctble_iff_equinum_set_nat] at h1\n rewrite [ctble_iff_equinum_set_nat] at h2\n obtain (I : Set Nat) (h3 : I \u223c A) from h1\n obtain (J : Set Nat) (h4 : J \u223c B) from h2\n have h5 : I \u00d7\u209b J \u223c A \u00d7\u209b B := Theorem_8_1_2_1 h3 h4\n have h6 : I \u00d7\u209b J \u2286 Univ (Nat \u00d7 Nat) := by\n fix p : Nat \u00d7 Nat\n assume h6 : p \u2208 I \u00d7\u209b J\n show p \u2208 Univ (Nat \u00d7 Nat) from elt_Univ p\n done\n have h7 : ctble (Univ (Nat \u00d7 Nat)) := by\n define --Goal : finite (Univ (Nat \u00d7 Nat)) \u2228 denum (Univ (Nat \u00d7 Nat))\n apply Or.inr\n rewrite [denum_def]\n show Univ Nat \u223c Univ (Nat \u00d7 Nat) from Theorem_8_1_3_2 NxN_equinum_N\n done\n have h8 : ctble (I \u00d7\u209b J) := Exercise_8_1_17 h6 h7\n show ctble (A \u00d7\u209b B) from ctble_of_equinum_ctble h5 h8\n done\n\nlemma Lemma_8_2_2_1 {U : Type} {F : Set (Set U)} {f : Set U \u2192 Rel Nat U}\n (h1 : ctble F) (h2 : \u2200 A \u2208 F, fcnl_onto_from_nat (f A) A) :\n ctble (\u22c3\u2080 F) := by\n rewrite [Theorem_8_1_5_2] at h1\n rewrite [Theorem_8_1_5_2]\n obtain (R : Rel Nat (Set U)) (h3 : fcnl_onto_from_nat R F) from h1\n define at h3\n have Runiqueval : unique_val_on_N R := h3.left\n have Ronto : nat_rel_onto R F := h3.right\n set S : Rel Nat U := enum_union_fam F f R\n apply Exists.intro S\n define\n apply And.intro\n \u00b7 -- Proof of unique_val_on_N S\n define\n fix n : Nat; fix a1 : U; fix a2 : U\n assume Sna1 : S n a1\n assume Sna2 : S n a2 --Goal : a1 = a2\n define at Sna1; define at Sna2\n obtain ((i1, j1) : Nat \u00d7 Nat) (h4 : fnnn (i1, j1) = n \u2227\n \u2203 A \u2208 F, R i1 A \u2227 f A j1 a1) from Sna1\n obtain (A1 : Set U) (Aija1 : A1 \u2208 F \u2227 R i1 A1 \u2227 f A1 j1 a1)\n from h4.right\n obtain ((i2, j2) : Nat \u00d7 Nat) (h5 : fnnn (i2, j2) = n \u2227\n \u2203 A \u2208 F, R i2 A \u2227 f A j2 a2) from Sna2\n obtain (A2 : Set U) (Aija2 : A2 \u2208 F \u2227 R i2 A2 \u2227 f A2 j2 a2)\n from h5.right\n rewrite [\u2190h5.left] at h4\n have h6 : (i1, j1) = (i2, j2) :=\n fnnn_one_one (i1, j1) (i2, j2) h4.left\n have h7 : i1 = i2 \u2227 j1 = j2 := Prod.mk.inj h6\n rewrite [h7.left, h7.right] at Aija1\n --Aija1 : A1 \u2208 F \u2227 R i2 A1 \u2227 f A1 j2 a1\n define at Runiqueval\n have h8 : A1 = A2 := Runiqueval Aija1.right.left Aija2.right.left\n rewrite [h8] at Aija1 --Aija1 : A2 \u2208 F \u2227 R i2 A2 \u2227 f A2 j2 a1\n have fA2fcnlonto : fcnl_onto_from_nat (f A2) A2 := h2 A2 Aija2.left\n define at fA2fcnlonto\n have fA2uniqueval : unique_val_on_N (f A2) := fA2fcnlonto.left\n define at fA2uniqueval\n show a1 = a2 from fA2uniqueval Aija1.right.right Aija2.right.right\n done\n \u00b7 -- Proof of nat_rel_onto S (\u22c3\u2080 F)\n define\n fix x : U\n assume h4 : x \u2208 \u22c3\u2080 F --Goal : \u2203 (n : Nat), S n x\n define at h4\n obtain (A : Set U) (h5 : A \u2208 F \u2227 x \u2208 A) from h4\n define at Ronto\n obtain (i : Nat) (h6 : R i A) from Ronto h5.left\n have fAfcnlonto : fcnl_onto_from_nat (f A) A := h2 A h5.left\n define at fAfcnlonto\n have fAonto : nat_rel_onto (f A) A := fAfcnlonto.right\n define at fAonto\n obtain (j : Nat) (h7 : f A j x) from fAonto h5.right\n apply Exists.intro (fnnn (i, j))\n define --Goal : \u2203 (p : Nat \u00d7 Nat), fnnn p = fnnn (i, j) \u2227\n -- \u2203 A \u2208 F, R p.1 A \u2227 f A p.2 x\n apply Exists.intro (i, j)\n apply And.intro\n \u00b7 -- Proof that fnnn (i, j) = fnnn (i, j)\n rfl\n done\n \u00b7 -- Proof that \u2203 A \u2208 F, R (i, j).1 A \u2227 f A (i, j).2 x\n apply Exists.intro A\n show A \u2208 F \u2227 R (i, j).1 A \u2227 f A (i, j).2 x from\n And.intro h5.left (And.intro h6 h7)\n done\n done\n done\n\nlemma Lemma_8_2_2_2 {U : Type} {F : Set (Set U)} (h : \u2200 A \u2208 F, ctble A) :\n \u2203 (f : Set U \u2192 Rel Nat U), \u2200 A \u2208 F, fcnl_onto_from_nat (f A) A := by\n have h1 : \u2200 (A : Set U), \u2203 (SA : Rel Nat U),\n A \u2208 F \u2192 fcnl_onto_from_nat SA A := by\n fix A : Set U\n by_cases h2 : A \u2208 F\n \u00b7 -- Case 1. h2 : A \u2208 F\n have h3 : ctble A := h A h2\n rewrite [Theorem_8_1_5_2] at h3\n obtain (SA : Rel Nat U) (h4 : fcnl_onto_from_nat SA A) from h3\n apply Exists.intro SA\n assume h5 : A \u2208 F\n show fcnl_onto_from_nat SA A from h4\n done\n \u00b7 -- Case 2. h2 : A \u2209 F\n apply Exists.intro (emptyRel Nat U)\n assume h3 : A \u2208 F\n show fcnl_onto_from_nat (emptyRel Nat U) A from absurd h3 h2\n done\n done\n set f : Set U \u2192 Rel Nat U := fun (A : Set U) => Classical.choose (h1 A)\n apply Exists.intro f\n fix A : Set U\n show A \u2208 F \u2192 fcnl_onto_from_nat (f A) A from Classical.choose_spec (h1 A)\n done\n\ntheorem Theorem_8_2_2 {U : Type} {F : Set (Set U)}\n (h1 : ctble F) (h2 : \u2200 A \u2208 F, ctble A) : ctble (\u22c3\u2080 F) := by\n obtain (f : Set U \u2192 Rel Nat U) (h3 : \u2200 A \u2208 F, fcnl_onto_from_nat (f A) A)\n from Lemma_8_2_2_2 h2\n show ctble 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"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.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.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "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.Meta.Tactic.Delta", "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.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "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.Meta.Tactic.Symm", "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.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "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", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6", "HTPILib.Chap7"]}, "proofMetadata": {"hasProof": true, "proof": ":= by rfl", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 9}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap4\nnamespace HTPI\n\n/- Definitions -/\ndef graph {A B : Type} (f : A \u2192 B) : Set (A \u00d7 B) :=\n {(a, b) : A \u00d7 B | f a = b}\n\ndef is_func_graph {A B : Type} (G : Set (A \u00d7 B)) : Prop :=\n \u2200 (x : A), \u2203! (y : B), (x, y) \u2208 G\n\ndef onto {A B : Type} (f : A \u2192 B) : Prop :=\n \u2200 (y : B), \u2203 (x : A), f x = y\n\ndef one_to_one {A B : Type} (f : A \u2192 B) : Prop :=\n \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n\ndef closed {A : Type} (f : A \u2192 A) (C : Set A) : Prop := \u2200 x \u2208 C, f x \u2208 C\n\ndef closure {A : Type} (f : A \u2192 A) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed f D}\n\ndef closed2 {A : Type} (f : A \u2192 A \u2192 A) (C : Set A) : Prop :=\n \u2200 x \u2208 C, \u2200 y \u2208 C, f x y \u2208 C\n\ndef closure2 {A : Type} (f : A \u2192 A \u2192 A) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed2 f D}\n\ndef closed_family {A : Type} (F : Set (A \u2192 A)) (C : Set A) : Prop :=\n \u2200 f \u2208 F, closed f C\n\ndef closure_family {A : Type} (F : Set (A \u2192 A)) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed_family F D}\n\ndef image {A B : Type} (f : A \u2192 B) (X : Set A) : Set B :=\n {f x | x \u2208 X}\n\ndef inverse_image {A B : Type} (f : A \u2192 B) (Y : Set B) : Set A :=\n {a : A | f a \u2208 Y}\n\n/- Section 5.1 -/\ntheorem graph_def {A B : Type} (f : A \u2192 B) (a : A) (b : B) :\n (a, b) \u2208 graph f \u2194 f a = b := by rfl\n\ntheorem func_from_graph_ltr {A B : Type} (F : Set (A \u00d7 B)) :\n (\u2203 (f : A \u2192 B), graph f = F) \u2192 is_func_graph F := sorry\n\n--This proof is explained in Section 8.2\ntheorem func_from_graph_rtl {A B : Type} (F : Set (A \u00d7 B)) :\n is_func_graph F \u2192 (\u2203 (f : A \u2192 B), graph f = F) := by\n assume h1 : is_func_graph F\n define at h1 --h1 : \u2200 (x : A), \u2203! (y : B), (x, y) \u2208 F\n have h2 : \u2200 (x : A), \u2203 (y : B), (x, y) \u2208 F := by\n fix x : A\n obtain (y : B) (h3 : (x, y) \u2208 F)\n (h4 : \u2200 (y1 y2 : B), (x, y1) \u2208 F \u2192 (x, y2) \u2208 F \u2192 y1 = y2) from h1 x\n show \u2203 (y : B), (x, y) \u2208 F from Exists.intro y h3\n done\n set f : A \u2192 B := fun (x : A) => Classical.choose (h2 x)\n apply Exists.intro f\n apply Set.ext\n fix (x, y) : A \u00d7 B\n have h3 : (x, f x) \u2208 F := Classical.choose_spec (h2 x)\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h4 : (x, y) \u2208 graph f\n define at h4 --h4 : f x = y\n rewrite [h4] at h3\n show (x, y) \u2208 F from h3\n done\n \u00b7 -- (\u2190)\n assume h4 : (x, y) \u2208 F\n define --Goal : f x = y\n obtain (z : B) (h5 : (x, z) \u2208 F)\n (h6 : \u2200 (y1 y2 : B), (x, y1) \u2208 F \u2192 (x, y2) \u2208 F \u2192 y1 = y2) from h1 x\n show f x = y from h6 (f x) y h3 h4\n done\n done\n\n", "theoremStatement": "theorem func_from_graph {A B : Type} (F : Set (A 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"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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n Iff.intro (func_from_graph_ltr F) (func_from_graph_rtl F)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 62}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap7\nnamespace HTPI\n\n/- Definitions -/\ndef fnz (n : Nat) : Int := if 2 \u2223 n then \u2191(n / 2) else -\u2191((n + 1) / 2)\n\ndef fzn (a : Int) : Nat := if a \u2265 0 then 2 * Int.toNat a else 2 * Int.toNat (-a) - 1\n\ndef tri (k : Nat) : Nat := k * (k + 1) / 2\n\ndef fnnn (p : Nat \u00d7 Nat) : Nat := tri (p.1 + p.2) + p.1\n\ndef num_elts_below (A : Set Nat) (m s : Nat) : Prop :=\n match m with\n | 0 => s = 0\n | n + 1 => (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s)\n\ndef enum (A : Set Nat) (s n : Nat) : Prop := n \u2208 A \u2227 num_elts_below A n s\n\ndef unique_val_on_N {U : Type} (R : Rel Nat U) : Prop :=\n \u2200 \u2983n : Nat\u2984 \u2983x1 x2 : U\u2984, R n x1 \u2192 R n x2 \u2192 x1 = x2\n\ndef nat_rel_onto {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203 (n : Nat), R n x\n\ndef fcnl_onto_from_nat {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n unique_val_on_N R \u2227 nat_rel_onto R A\n\ndef fcnl_one_one_to_nat {U : Type} (R : Rel U Nat) (A : Set U) : Prop :=\n fcnl_on R A \u2227 \u2200 \u2983x1 x2 : U\u2984 \u2983n : Nat\u2984,\n (x1 \u2208 A \u2227 R x1 n) \u2192 (x2 \u2208 A \u2227 R x2 n) \u2192 x1 = x2\n\ndef least_rel_to {U : Type} (S : Rel Nat U) (x : U) (n : Nat) : Prop :=\n S n x \u2227 \u2200 (m : Nat), S m x \u2192 n \u2264 m\n\ndef restrict_to {U V : Type} (S : Rel U V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 S x y\n\ndef fqn (q : Rat) : Nat := fnnn (fzn q.num, q.den)\n\ndef Set_rp_below (m : Nat) : Set Nat := {n : Nat | rel_prime m n \u2227 n < m}\n\ndef Set_prod {U V : Type} (A : Set U) (B : Set V) : Set (U \u00d7 V) :=\n {(a, b) : U \u00d7 V | a \u2208 A \u2227 b \u2208 B}\n\nnotation:75 A:75 \" \u00d7\u209b \" B:75 => Set_prod A B\n\ndef Rel_prod {U V W X : Type} (R : Rel U V) (S : Rel W X)\n (p : U \u00d7 W) (q : V \u00d7 X) : Prop := R p.1 q.1 \u2227 S p.2 q.2\n\nnotation:75 R:75 \" \u00d7\u1d63 \" S:75 => Rel_prod R S\n\ndef qr (n a : Nat) : Nat \u00d7 Nat := (a / n, a % n)\n\ndef mod_mod (m n a : Nat) : Nat \u00d7 Nat := (a % m, a % n)\n\ndef enum_union_fam {U : Type}\n (F : Set (Set U)) (f : Set U \u2192 Rel Nat U) (R : Rel Nat (Set U))\n (n : Nat) (a : U) : Prop := \u2203 (p : Nat \u00d7 Nat), fnnn p = n \u2227\n \u2203 A \u2208 F, R p.1 A \u2227 (f A) p.2 a\n\ndef seq {U : Type} (A : Set U) : Set (List U) :=\n {l : List U | \u2200 x \u2208 l, x \u2208 A}\n\ndef seq_by_length {U : Type} (A : Set U) (n : Nat) : Set (List U) :=\n {l : List U | l \u2208 seq A \u2227 l.length = n}\n\ndef seq_cons (U : Type) (p : U \u00d7 (List U)) : List U := p.1 :: p.2\n\ndef sbl_set {U : Type} (A : Set U) : Set (Set (List U)) :=\n {S : Set (List U) | \u2203 (n : Nat), seq_by_length A n = S}\n\ndef rep_common_image\n {U V : Type} (R S : Rel U V) (X0 : Set U) (n : Nat) : Set U :=\n match n with\n | 0 => X0\n | m + 1 => {a : U | \u2203 x \u2208 rep_common_image R S X0 m,\n \u2203 (y : V), R x y \u2227 S a y}\n\ndef cum_rep_image {U V : Type} (R S : Rel U V) (X0 : Set U) : Set U :=\n {a : U | \u2203 (n : Nat), a \u2208 rep_common_image R S X0 n}\n\ndef csb_match {U V : Type} (R S : Rel U V) (X0 : Set U)\n (x : U) (y : V) : Prop := x \u2208 cum_rep_image R S X0 \u2227 R x y \u2228\n x \u2209 cum_rep_image R S X0 \u2227 S x y\n\n/- Section 8.1 -/\n#eval [fnz 0, fnz 1, fnz 2, fnz 3, fnz 4, fnz 5, fnz 6]\n --Answer: [0, -1, 1, -2, 2, -3, 3]\n\n#eval [fzn 0, fzn (-1), fzn 1, fzn (-2), fzn 2, fzn (-3), fzn 3]\n --Answer: [0, 1, 2, 3, 4, 5, 6]\n\nlemma fnz_even (k : Nat) : fnz (2 * k) = \u2191k := by\n have h1 : 2 \u2223 2 * k := by\n apply Exists.intro k\n rfl\n done\n have h2 : fnz (2 * k) = if 2 \u2223 2 * k then \u2191(2 * k / 2)\n else -\u2191((2 * k + 1) / 2) := by rfl\n rewrite [if_pos h1] at h2 --h2 : fnz (2 * k) = \u2191(2 * k / 2)\n have h3 : 0 < 2 := by linarith\n rewrite [Nat.mul_div_cancel_left k h3] at h2\n show fnz (2 * k) = \u2191k from h2\n done\n\nlemma fnz_odd (k : Nat) : fnz (2 * k + 1) = -\u2191(k + 1) := sorry\n\nlemma fzn_nat (k : Nat) : fzn \u2191k = 2 * k := by rfl\n\nlemma fzn_neg_succ_nat (k : Nat) : fzn (-\u2191(k + 1)) = 2 * k + 1 := by rfl\n\n--From exercises of Section 6.1\ntheorem Exercise_6_1_16a1 : \u2200 (n : Nat), nat_even n \u2228 nat_odd n := sorry\n\nlemma fzn_fnz : fzn \u2218 fnz = id := by\n apply funext --Goal : \u2200 (x : Nat), (fzn \u2218 fnz) x = id x\n fix n : Nat\n rewrite [comp_def] --Goal : fzn (fnz n) = id n\n have h1 : nat_even n \u2228 nat_odd n := Exercise_6_1_16a1 n\n by_cases on h1\n \u00b7 -- Case 1. h1 : nat_even n\n obtain (k : Nat) (h2 : n = 2 * k) from h1\n rewrite [h2, fnz_even, fzn_nat]\n rfl\n done\n \u00b7 -- Case 2. h1 : nat_odd n\n obtain (k : Nat) (h2 : n = 2 * k + 1) from h1\n rewrite [h2, fnz_odd, fzn_neg_succ_nat]\n rfl\n done\n done\n\nlemma fnz_fzn : fnz \u2218 fzn = id := sorry\n\nlemma fzn_one_one : one_to_one fzn := Theorem_5_3_3_1 fzn fnz fnz_fzn\n\nlemma fzn_onto : onto fzn := Theorem_5_3_3_2 fzn fnz fzn_fnz\n\nlemma fnz_one_one : one_to_one fnz := Theorem_5_3_3_1 fnz fzn fzn_fnz\n\nlemma fnz_onto : onto fnz := Theorem_5_3_3_2 fnz fzn fnz_fzn\n\nlemma fnnn_def (a b : Nat) : fnnn (a, b) = tri (a + b) + a := by rfl\n\n#eval [fnnn (0, 0), fnnn (0, 1), fnnn (1, 0), fnnn (0, 2), fnnn (1, 1)]\n --Answer: [0, 1, 2, 3, 4]\n\nlemma tri_step (k : Nat) : tri (k + 1) = tri k + k + 1 := sorry\n\nlemma tri_incr {j k : Nat} (h1 : j \u2264 k) : tri j \u2264 tri k := sorry\n\nlemma le_of_fnnn_eq {a1 b1 a2 b2 : Nat}\n (h1 : fnnn (a1, b1) = fnnn (a2, b2)) : a1 + b1 \u2264 a2 + b2 := by\n by_contra h2\n have h3 : a2 + b2 + 1 \u2264 a1 + b1 := by linarith\n have h4 : fnnn (a2, b2) < fnnn (a1, b1) :=\n calc fnnn (a2, b2)\n _ = tri (a2 + b2) + a2 := by rfl\n _ < tri (a2 + b2) + (a2 + b2) + 1 := by linarith\n _ = tri (a2 + b2 + 1) := (tri_step _).symm\n _ \u2264 tri (a1 + b1) := tri_incr h3\n _ \u2264 tri (a1 + b1) + a1 := by linarith\n _ = fnnn (a1, b1) := by rfl\n linarith\n done\n\nlemma fnnn_one_one : one_to_one fnnn := by\n fix (a1, b1) : Nat \u00d7 Nat\n fix (a2, b2) : Nat \u00d7 Nat\n assume h1 : fnnn (a1, b1) = fnnn (a2, b2) --Goal : (a1, b1) = (a2, b2)\n have h2 : a1 + b1 \u2264 a2 + b2 := le_of_fnnn_eq h1\n have h3 : a2 + b2 \u2264 a1 + b1 := le_of_fnnn_eq h1.symm\n have h4 : a1 + b1 = a2 + b2 := by linarith\n rewrite [fnnn_def, fnnn_def, h4] at h1\n --h1 : tri (a2 + b2) + a1 = tri (a2 + b2) + a2\n have h6 : a1 = a2 := Nat.add_left_cancel h1\n rewrite [h6] at h4 --h4 : a2 + b1 = a2 + b2\n have h7 : b1 = b2 := Nat.add_left_cancel h4\n rewrite [h6, h7]\n rfl\n done\n\nlemma fnnn_onto : onto fnnn := by\n define --Goal : \u2200 (y : Nat), \u2203 (x : Nat \u00d7 Nat), fnnn x = y\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro (0, 0)\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2203 (x : Nat \u00d7 Nat), fnnn x = n\n obtain ((a, b) : Nat \u00d7 Nat) (h1 : fnnn (a, b) = n) from ih\n by_cases h2 : b = 0\n \u00b7 -- Case 1. h2 : b = 0\n apply Exists.intro (0, a + 1)\n show fnnn (0, a + 1) = n + 1 from\n calc fnnn (0, a + 1)\n _ = tri (0 + (a + 1)) + 0 := by rfl\n _ = tri (a + 1) := by ring\n _ = tri a + a + 1 := tri_step a\n _ = tri (a + 0) + a + 1 := by ring\n _ = fnnn (a, b) + 1 := by rw [h2, fnnn_def]\n _ = n + 1 := by rw [h1]\n done\n \u00b7 -- Case 2. h2 : b \u2260 0\n obtain (k : Nat) (h3 : b = k + 1) from\n exists_eq_add_one_of_ne_zero h2\n apply Exists.intro (a + 1, k)\n show fnnn (a + 1, k) = n + 1 from\n calc fnnn (a + 1, k)\n _ = tri (a + 1 + k) + (a + 1) := by rfl\n _ = tri (a + (k + 1)) + a + 1 := by ring\n _ = tri (a + b) + a + 1 := by rw [h3]\n _ = fnnn (a, b) + 1 := by rfl\n _ = n + 1 := by rw [h1]\n done\n done\n done\n\nlemma one_one_on_of_one_one {U V : Type} {f : U \u2192 V}\n (h : one_to_one f) (A : Set U) : one_one_on f A := by\n define\n fix x1 : U; fix x2 : U\n assume h1 : x1 \u2208 A\n assume h2 : x2 \u2208 A\n show f x1 = f x2 \u2192 x1 = x2 from h x1 x2\n done\n\nlemma elt_Univ {U : Type} (u : U) :\n u \u2208 Univ U := by trivial\n\ntheorem equinum_Univ {U V : Type} {f : U \u2192 V}\n (h1 : one_to_one f) (h2 : onto f) : Univ U \u223c Univ V := by\n have h3 : image f (Univ U) = Univ V := by\n apply Set.ext\n fix v : V\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h3 : v \u2208 image f (Univ U)\n show v \u2208 Univ V from elt_Univ v\n done\n \u00b7 -- (\u2190)\n assume h3 : v \u2208 Univ V\n obtain (u : U) (h4 : f u = v) from h2 v\n apply Exists.intro u\n apply And.intro _ h4\n show u \u2208 Univ U from elt_Univ u\n done\n done\n show Univ U \u223c Univ V from\n equinum_image (one_one_on_of_one_one h1 (Univ U)) h3\n done\n\ntheorem Z_equinum_N : Univ Int \u223c Univ Nat :=\n equinum_Univ fzn_one_one fzn_onto\n\ntheorem NxN_equinum_N : Univ (Nat \u00d7 Nat) \u223c Univ Nat :=\n equinum_Univ fnnn_one_one fnnn_onto\n\nlemma neb_step (A : Set Nat) (n s : Nat) : num_elts_below A (n + 1) s \u2194\n (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s) := by rfl\n\nlemma neb_step_elt {A : Set Nat} {n : Nat} (h1 : n \u2208 A) (s : Nat) :\n num_elts_below A (n + 1) s \u2194 1 \u2264 s \u2227 num_elts_below A n (s - 1) := by\n rewrite [neb_step]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) \u2228\n \u00acn \u2208 A \u2227 num_elts_below A n s\n by_cases on h2\n \u00b7 -- Case 1. h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n show 1 \u2264 s \u2227 num_elts_below A n (s - 1) from h2.right\n done\n \u00b7 -- Case 2. h2 : \u00acn \u2208 A \u2227 num_elts_below A n s\n show 1 \u2264 s \u2227 num_elts_below A n (s - 1) from absurd h1 h2.left\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n apply Or.inl\n show n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) from And.intro h1 h2\n done\n done\n\nlemma neb_step_not_elt {A : Set Nat} {n : Nat} (h1 : n \u2209 A) (s : Nat) :\n num_elts_below A (n + 1) s \u2194 num_elts_below A n s := by\n rewrite [neb_step]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) \u2228\n \u00acn \u2208 A \u2227 num_elts_below A n s\n by_cases on h2\n \u00b7 -- Case 1. h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n show num_elts_below A n s from absurd h2.left h1\n done\n \u00b7 -- Case 2. h2 : \u00acn \u2208 A \u2227 num_elts_below A n s\n show num_elts_below A n s from h2.right\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : num_elts_below A n s\n apply Or.inr\n show \u00acn \u2208 A \u2227 num_elts_below A n s from And.intro h1 h2\n done\n done\n\nlemma neb_exists (A : Set Nat) :\n \u2200 (n : Nat), \u2203 (s : Nat), num_elts_below A n s := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro 0\n define\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2203 (s : Nat), num_elts_below A n s\n obtain (t : Nat) (h1 : num_elts_below A n t) from ih\n by_cases h2 : n \u2208 A\n \u00b7 -- Case 1. h2 : n \u2208 A\n apply Exists.intro (t + 1)\n rewrite [neb_step_elt h2, Nat.add_sub_cancel]\n apply And.intro _ h1\n linarith\n done\n \u00b7 -- Case 2. h2 : n \u2209 A\n apply Exists.intro t\n rewrite [neb_step_not_elt h2]\n show num_elts_below A n t from h1\n done\n done\n done\n\nlemma neb_unique (A : Set Nat) : \u2200 \u2983n : Nat\u2984, \u2200 \u2983s1 s2 : Nat\u2984,\n num_elts_below A n s1 \u2192 num_elts_below A n s2 \u2192 s1 = s2 := by\n by_induc\n \u00b7 -- Base Case\n fix s1 : Nat; fix s2 : Nat\n assume h1 : num_elts_below A 0 s1\n assume h2 : num_elts_below A 0 s2\n define at h1; define at h2 --h1 : s1 = 0; h2 : s2 = 0\n rewrite [h1, h2]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 \u2983s1 s2 : Nat\u2984,\n num_elts_below A n s1 \u2192 num_elts_below A n s2 \u2192 s1 = s2\n fix s1 : Nat; fix s2 : Nat\n assume h1 : num_elts_below A (n + 1) s1\n assume h2 : num_elts_below A (n + 1) s2\n by_cases h3 : n \u2208 A\n \u00b7 -- Case 1. h3 : n \u2208 A\n rewrite [neb_step_elt h3] at h1\n rewrite [neb_step_elt h3] at h2\n --h1 : 1 \u2264 s1 \u2227 num_elts_below A n (s1 - 1)\n --h2 : 1 \u2264 s2 \u2227 num_elts_below A n (s2 - 1)\n have h4 : s1 - 1 = s2 - 1 := ih h1.right h2.right\n show s1 = s2 from\n calc s1\n _ = s1 - 1 + 1 := (Nat.sub_add_cancel h1.left).symm\n _ = s2 - 1 + 1 := by rw [h4]\n _ = s2 := Nat.sub_add_cancel h2.left\n done\n \u00b7 -- Case 2. h3 : n \u2209 A\n rewrite [neb_step_not_elt h3] at h1 --h1 : num_elts_below A n s1\n rewrite [neb_step_not_elt h3] at h2 --h2 : num_elts_below A n s2\n show s1 = s2 from ih h1 h2\n done\n done\n done\n\nlemma neb_increase {A : Set Nat} {n s : Nat} (h1 : enum A s n) :\n \u2200 \u2983m : Nat\u2984, m \u2265 n + 1 \u2192 \u2200 \u2983t : Nat\u2984, num_elts_below A m t \u2192 s < t := by\n by_induc\n \u00b7 -- Base Case\n define at h1\n fix t : Nat\n assume h2 : num_elts_below A (n + 1) t\n rewrite [neb_step_elt h1.left] at h2\n have h3 : s = t - 1 := neb_unique A h1.right h2.right\n show s < t from\n calc s\n _ = t - 1 := h3\n _ < t - 1 + 1 := by linarith\n _ = t := Nat.sub_add_cancel h2.left\n done\n \u00b7 -- Induction Step\n fix m : Nat\n assume h2 : m \u2265 n + 1\n assume ih : \u2200 \u2983t : Nat\u2984, num_elts_below A m t \u2192 s < t\n fix t : Nat\n assume h3 : num_elts_below A (m + 1) t\n by_cases h4 : m \u2208 A\n \u00b7 -- Case 1. h4 : m \u2208 A\n rewrite [neb_step_elt h4] at h3\n have h5 : s < t - 1 := ih h3.right\n show s < t from\n calc s\n _ < t - 1 := h5\n _ \u2264 t := Nat.sub_le _ _\n done\n \u00b7 -- Case 2. h4 : m \u2209 A\n rewrite [neb_step_not_elt h4] at h3\n show s < t from ih h3\n done\n done\n done\n\nlemma enum_not_skip {A : Set Nat} : \u2200 \u2983m s : Nat\u2984, num_elts_below A m s \u2192\n \u2200 t < s, \u2203 (n : Nat), enum A t n := by\n by_induc\n \u00b7 -- Base Case\n fix s : Nat\n assume h1 : num_elts_below A 0 s\n define at h1\n fix t : Nat\n contrapos\n assume h2 : \u00ac\u2203 (n : Nat), enum A t n\n linarith\n done\n \u00b7 -- Induction Step\n fix m : Nat\n assume ih : \u2200 \u2983s : Nat\u2984, num_elts_below A m s \u2192 \u2200 t < s, \u2203 (n : Nat), enum A t n\n fix s : Nat\n assume h1 : num_elts_below A (m + 1) s\n by_cases h2 : m \u2208 A\n \u00b7 -- Case 1. h2 : m \u2208 A\n rewrite [neb_step_elt h2] at h1\n have h3 : \u2200 t < s - 1, \u2203 (n : Nat), enum A t n := ih h1.right\n fix t : Nat\n assume h4 : t < s\n by_cases h5 : t = s - 1\n \u00b7 -- Case 1.1. h5 : t = s - 1\n apply Exists.intro m\n define\n apply And.intro h2\n rewrite [h5]\n show num_elts_below A m (s - 1) from h1.right\n done\n \u00b7 -- Case 1.2. h5 : t \u2260 s - 1\n have h6 : t \u2264 s - 1 := Nat.le_pred_of_lt h4\n have h7 : t < s - 1 := Nat.lt_of_le_of_ne h6 h5\n show \u2203 (n : Nat), enum A t n from ih h1.right t h7\n done\n done\n \u00b7 -- Case 2. h2 : m \u2209 A\n rewrite [neb_step_not_elt h2] at h1\n show \u2200 t < s, \u2203 (n : Nat), enum A t n from ih h1\n done\n done\n done\n\nlemma enum_le {A : Set Nat} {t n1 n2 : Nat}\n (h1 : enum A t n1) (h2 : enum A t n2) : n1 \u2264 n2 := by\n by_contra h3\n have h4 : n2 + 1 \u2264 n1 := by linarith\n define at h1\n have h5 : t < t := neb_increase h2 h4 h1.right\n linarith\n done\n\nlemma enum_unique (A : Set Nat) (t : Nat) :\n \u2200 \u2983n1 n2 : Nat\u2984, enum A t n1 \u2192 enum A t n2 \u2192 n1 = n2 := by\n fix n1 : Nat; fix n2 : Nat\n assume h1 : enum A t n1\n assume h2 : enum A t n2\n have h3 : n1 \u2264 n2 := enum_le h1 h2\n have h4 : n2 \u2264 n1 := enum_le h2 h1\n linarith\n done\n\nlemma inv_enum_fcnl (A : Set Nat) : fcnl_on (invRel (enum A)) A := by\n define\n fix n : Nat\n assume h1 : n \u2208 A\n exists_unique\n \u00b7 -- Existence\n obtain (s : Nat) (h2 : num_elts_below A n s) from neb_exists A n\n apply Exists.intro s\n define\n show n \u2208 A \u2227 num_elts_below A n s from And.intro h1 h2\n done\n \u00b7 -- Uniqueness\n fix s1 : Nat; fix s2 : Nat\n assume h2 : invRel (enum A) n s1\n assume h3 : invRel (enum A) n s2\n define at h2; define at h3\n show s1 = s2 from neb_unique A h2.right h3.right\n done\n done\n\nlemma bdd_subset_nat_match {A : Set Nat} {m s : Nat}\n (h1 : \u2200 n \u2208 A, n < m) (h2 : num_elts_below A m s) :\n matching (enum A) (I s) A := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix t : Nat; fix n : Nat\n assume h3 : enum A t n\n have h4 : \u2200 \u2983m : Nat\u2984, m \u2265 n + 1 \u2192 \u2200 \u2983t_1 : Nat\u2984, num_elts_below A m t_1 \u2192 t < t_1 :=\n neb_increase h3\n define at h3\n apply And.intro _ h3.left\n define\n have h5 : n < m := h1 n h3.left\n have h6 : m \u2265 n + 1 := h5\n show t < s from h4 h6 h2\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- proof of fcnl_on (enum A) (I s)\n define\n fix t : Nat\n assume h3 : t \u2208 I s\n define at h3\n exists_unique\n \u00b7 -- Existence\n show \u2203 (y : Nat), enum A t y from enum_not_skip h2 t h3\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A t y_1 \u2192 enum A t y_2 \u2192 y_1 = y_2 from\n enum_unique A t\n done\n done\n \u00b7 -- Proof of fcnl_on (invRel (enum A)) A\n show fcnl_on (invRel (enum A)) A from inv_enum_fcnl A\n done\n done\n done\n\nlemma bdd_subset_nat {A : Set Nat} {m s : Nat}\n (h1 : \u2200 n \u2208 A, n < m) (h2 : num_elts_below A m s) :\n I s \u223c A := Exists.intro (enum A) (bdd_subset_nat_match h1 h2)\n\nlemma enum_fcnl_of_unbdd {A : Set Nat} (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n fcnl_on (enum A) (Univ Nat) := by\n define\n by_induc\n \u00b7 -- Base Case\n assume h2 : 0 \u2208 Univ Nat\n exists_unique\n \u00b7 -- Existence\n obtain (n : Nat) (h3 : n \u2208 A \u2227 n \u2265 0) from h1 0\n obtain (s : Nat) (h4 : num_elts_below A (n + 1) s) from neb_exists A (n + 1)\n have h5 : \u2200 (t : Nat), t < s \u2192 \u2203 (n : Nat), enum A t n := enum_not_skip h4\n rewrite [neb_step_elt h3.left] at h4\n show \u2203 (y : Nat), enum A 0 y from h5 0 h4.left\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A 0 y_1 \u2192 enum A 0 y_2 \u2192 y_1 = y_2 from enum_unique A 0\n done\n done\n \u00b7 -- Induction Step\n fix s : Nat\n assume ih : s \u2208 Univ Nat \u2192 \u2203! (y : Nat), enum A s y\n assume h2 : s + 1 \u2208 Univ Nat\n exists_unique\n \u00b7 -- Existence\n have h3 : s \u2208 Univ Nat := elt_Univ s\n obtain (m : Nat) (h4 : enum A s m)\n (h5 : \u2200 (y_1 y_2 : Nat), enum A s y_1 \u2192 enum A s y_2 \u2192 y_1 = y_2) from ih h3\n obtain (n : Nat) (h6 : n \u2208 A \u2227 n \u2265 m + 1) from h1 (m + 1)\n obtain (t : Nat) (h7 : num_elts_below A n t) from neb_exists A n\n have h8 : s < t := neb_increase h4 h6.right h7\n have h9 : s + 1 < t \u2228 s + 1 = t := Nat.lt_or_eq_of_le h8\n by_cases on h9\n \u00b7 -- Case 1. h9 : s + 1 < t\n show \u2203 (y : Nat), enum A (s + 1) y from enum_not_skip h7 (s + 1) h9\n done\n \u00b7 -- Case 2. h9 : s + 1 = t\n rewrite [h9]\n apply Exists.intro n\n define\n show n \u2208 A \u2227 num_elts_below A n t from And.intro h6.left h7\n done\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A (s + 1) y_1 \u2192 enum A (s + 1) y_2 \u2192 y_1 = y_2\n from enum_unique A (s + 1)\n done\n done\n done\n\nlemma unbdd_subset_nat_match {A : Set Nat}\n (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n matching (enum A) (Univ Nat) A := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix s : Nat; fix n : Nat\n assume h2 : enum A s n\n define at h2\n apply And.intro (elt_Univ s) h2.left\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on (enum A)\n show fcnl_on (enum A) (Univ Nat) from enum_fcnl_of_unbdd h1\n done\n \u00b7 -- Proof of fcnl_on (invRel (enum A))\n show fcnl_on (invRel (enum A)) A from inv_enum_fcnl A\n done\n done\n done\n\nlemma unbdd_subset_nat {A : Set Nat}\n (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n denum A := Exists.intro (enum A) (unbdd_subset_nat_match h1)\n\n", "theoremStatement": "lemma subset_nat_ctble (A : Set Nat) : ctble A ", "theoremName": "HTPI.subset_nat_ctble", "fileCreated": {"commit": "fe2721d65e0771872fe4cb8117517e643ac1dc1f", "date": "2023-05-10"}, "theoremCreated": 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"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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6", "HTPILib.Chap7"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n define --Goal : finite A \u2228 denum A\n by_cases h1 : \u2203 (m : Nat), \u2200 n \u2208 A, n < m\n \u00b7 -- Case 1. h1 : \u2203 (m : Nat), \u2200 n \u2208 A, n < m\n apply Or.inl --Goal : finite A\n obtain (m : Nat) (h2 : \u2200 n \u2208 A, n < m) from h1\n obtain (s : Nat) (h3 : num_elts_below A m s) from neb_exists A m\n apply Exists.intro s\n show I s \u223c A from bdd_subset_nat h2 h3\n done\n \u00b7 -- Case 2. h1 : \u00ac\u2203 (m : Nat), \u2200 n \u2208 A, n < m\n apply Or.inr --Goal : denum A\n push_neg at h1\n --This tactic converts h1 to \u2200 (m : Nat), \u2203 n \u2208 A, m \u2264 n\n show denum A from unbdd_subset_nat h1\n done\n done", "proofType": "tactic", "proofLengthLines": 16, "proofLengthTokens": 600}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\n", "theoremStatement": "lemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q ", "theoremName": "HTPI.exists_least_prime_factor", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "68d6f9e1b2d91bbfb4b31b37b871147a2eb4b63e", "date": "2023-04-28"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 309, "tokenPositionInFile": 9258, "theoremPositionInFile": 44}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 15, "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", 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"Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 226}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap7\nnamespace HTPI\n\n/- Definitions -/\ndef fnz (n : Nat) : Int := if 2 \u2223 n then \u2191(n / 2) else -\u2191((n + 1) / 2)\n\ndef fzn (a : Int) : Nat := if a \u2265 0 then 2 * Int.toNat a else 2 * Int.toNat (-a) - 1\n\ndef tri (k : Nat) : Nat := k * (k + 1) / 2\n\ndef fnnn (p : Nat \u00d7 Nat) : Nat := tri (p.1 + p.2) + p.1\n\ndef num_elts_below (A : Set Nat) (m s : Nat) : Prop :=\n match m with\n | 0 => s = 0\n | n + 1 => (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s)\n\ndef enum (A : Set Nat) (s n : Nat) : Prop := n \u2208 A \u2227 num_elts_below A n s\n\ndef unique_val_on_N {U : Type} (R : Rel Nat U) : Prop :=\n \u2200 \u2983n : Nat\u2984 \u2983x1 x2 : U\u2984, R n x1 \u2192 R n x2 \u2192 x1 = x2\n\ndef nat_rel_onto {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203 (n : Nat), R n x\n\ndef fcnl_onto_from_nat {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n unique_val_on_N R \u2227 nat_rel_onto R A\n\ndef fcnl_one_one_to_nat {U : Type} (R : Rel U Nat) (A : Set U) : Prop :=\n fcnl_on R A \u2227 \u2200 \u2983x1 x2 : U\u2984 \u2983n : Nat\u2984,\n (x1 \u2208 A \u2227 R x1 n) \u2192 (x2 \u2208 A \u2227 R x2 n) \u2192 x1 = x2\n\ndef least_rel_to {U : Type} (S : Rel Nat U) (x : U) (n : Nat) : Prop :=\n S n x \u2227 \u2200 (m : Nat), S m x \u2192 n \u2264 m\n\ndef restrict_to {U V : Type} (S : Rel U V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 S x y\n\ndef fqn (q : Rat) : Nat := fnnn (fzn q.num, q.den)\n\ndef Set_rp_below (m : Nat) : Set Nat := {n : Nat | rel_prime m n \u2227 n < m}\n\ndef Set_prod {U V : Type} (A : Set U) (B : Set V) : Set (U \u00d7 V) :=\n {(a, b) : U \u00d7 V | a \u2208 A \u2227 b \u2208 B}\n\nnotation:75 A:75 \" \u00d7\u209b \" B:75 => Set_prod A B\n\ndef Rel_prod {U V W X : Type} (R : Rel U V) (S : Rel W X)\n (p : U \u00d7 W) (q : V \u00d7 X) : Prop := R p.1 q.1 \u2227 S p.2 q.2\n\nnotation:75 R:75 \" \u00d7\u1d63 \" S:75 => Rel_prod R S\n\ndef qr (n a : Nat) : Nat \u00d7 Nat := (a / n, a % n)\n\ndef mod_mod (m n a : Nat) : Nat \u00d7 Nat := (a % m, a % n)\n\ndef enum_union_fam {U : Type}\n (F : Set (Set U)) (f : Set U \u2192 Rel Nat U) (R : Rel Nat (Set U))\n (n : Nat) (a : U) : Prop := \u2203 (p : Nat \u00d7 Nat), fnnn p = n \u2227\n \u2203 A \u2208 F, R p.1 A \u2227 (f A) p.2 a\n\ndef seq {U : Type} (A : Set U) : Set (List U) :=\n {l : List U | \u2200 x \u2208 l, x \u2208 A}\n\ndef seq_by_length {U : Type} (A : Set U) (n : Nat) : Set (List U) :=\n {l : List U | l \u2208 seq A \u2227 l.length = n}\n\ndef seq_cons (U : Type) (p : U \u00d7 (List U)) : List U := p.1 :: p.2\n\ndef sbl_set {U : Type} (A : Set U) : Set (Set (List U)) :=\n {S : Set (List U) | \u2203 (n : Nat), seq_by_length A n = S}\n\ndef rep_common_image\n {U V : Type} (R S : Rel U V) (X0 : Set U) (n : Nat) : Set U :=\n match n with\n | 0 => X0\n | m + 1 => {a : U | \u2203 x \u2208 rep_common_image R S X0 m,\n \u2203 (y : V), R x y \u2227 S a y}\n\ndef cum_rep_image {U V : Type} (R S : Rel U V) (X0 : Set U) : Set U :=\n {a : U | \u2203 (n : Nat), a \u2208 rep_common_image R S X0 n}\n\ndef csb_match {U V : Type} (R S : Rel U V) (X0 : Set U)\n (x : U) (y : V) : Prop := x \u2208 cum_rep_image R S X0 \u2227 R x y \u2228\n x \u2209 cum_rep_image R S X0 \u2227 S x y\n\n/- Section 8.1 -/\n#eval [fnz 0, fnz 1, fnz 2, fnz 3, fnz 4, fnz 5, fnz 6]\n --Answer: [0, -1, 1, -2, 2, -3, 3]\n\n#eval [fzn 0, fzn (-1), fzn 1, fzn (-2), fzn 2, fzn (-3), fzn 3]\n --Answer: [0, 1, 2, 3, 4, 5, 6]\n\nlemma fnz_even (k : Nat) : fnz (2 * k) = \u2191k := by\n have h1 : 2 \u2223 2 * k := by\n apply Exists.intro k\n rfl\n done\n have h2 : fnz (2 * k) = if 2 \u2223 2 * k then \u2191(2 * k / 2)\n else -\u2191((2 * k + 1) / 2) := by rfl\n rewrite [if_pos h1] at h2 --h2 : fnz (2 * k) = \u2191(2 * k / 2)\n have h3 : 0 < 2 := by linarith\n rewrite [Nat.mul_div_cancel_left k h3] at h2\n show fnz (2 * k) = \u2191k from h2\n done\n\nlemma fnz_odd (k : Nat) : fnz (2 * k + 1) = -\u2191(k + 1) := sorry\n\nlemma fzn_nat (k : Nat) : fzn \u2191k = 2 * k := by rfl\n\nlemma fzn_neg_succ_nat (k : Nat) : fzn (-\u2191(k + 1)) = 2 * k + 1 := by rfl\n\n--From exercises of Section 6.1\ntheorem Exercise_6_1_16a1 : \u2200 (n : Nat), nat_even n \u2228 nat_odd n := sorry\n\nlemma fzn_fnz : fzn \u2218 fnz = id := by\n apply funext --Goal : \u2200 (x : Nat), (fzn \u2218 fnz) x = id x\n fix n : Nat\n rewrite [comp_def] --Goal : fzn (fnz n) = id n\n have h1 : nat_even n \u2228 nat_odd n := Exercise_6_1_16a1 n\n by_cases on h1\n \u00b7 -- Case 1. h1 : nat_even n\n obtain (k : Nat) (h2 : n = 2 * k) from h1\n rewrite [h2, fnz_even, fzn_nat]\n rfl\n done\n \u00b7 -- Case 2. h1 : nat_odd n\n obtain (k : Nat) (h2 : n = 2 * k + 1) from h1\n rewrite [h2, fnz_odd, fzn_neg_succ_nat]\n rfl\n done\n done\n\nlemma fnz_fzn : fnz \u2218 fzn = id := sorry\n\nlemma fzn_one_one : one_to_one fzn := Theorem_5_3_3_1 fzn fnz fnz_fzn\n\nlemma fzn_onto : onto fzn := Theorem_5_3_3_2 fzn fnz fzn_fnz\n\nlemma fnz_one_one : one_to_one fnz := Theorem_5_3_3_1 fnz fzn fzn_fnz\n\nlemma fnz_onto : onto fnz := Theorem_5_3_3_2 fnz fzn fnz_fzn\n\nlemma fnnn_def (a b : Nat) : fnnn (a, b) = tri (a + b) + a := by rfl\n\n#eval [fnnn (0, 0), fnnn (0, 1), fnnn (1, 0), fnnn (0, 2), fnnn (1, 1)]\n --Answer: [0, 1, 2, 3, 4]\n\nlemma tri_step (k : Nat) : tri (k + 1) = tri k + k + 1 := sorry\n\nlemma tri_incr {j k : Nat} (h1 : j \u2264 k) : tri j \u2264 tri k := sorry\n\n", "theoremStatement": "lemma le_of_fnnn_eq {a1 b1 a2 b2 : Nat}\n (h1 : fnnn (a1, b1) = fnnn (a2, b2)) : a1 + b1 \u2264 a2 + b2 ", "theoremName": "HTPI.le_of_fnnn_eq", "fileCreated": {"commit": "fe2721d65e0771872fe4cb8117517e643ac1dc1f", "date": "2023-05-10"}, "theoremCreated": {"commit": "01538f73db4cc69cc47bd51e795bf25d251c4b6d", "date": "2023-05-31"}, "file": "htpi/HTPILib/Chap8Part2.lean", "module": "HTPILib.Chap8Part2", "jsonFile": "HTPILib.Chap8Part2.jsonl", 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"Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6", "HTPILib.Chap7"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n by_contra h2\n have h3 : a2 + b2 + 1 \u2264 a1 + b1 := by linarith\n have h4 : fnnn (a2, b2) < fnnn (a1, b1) :=\n calc fnnn (a2, b2)\n _ = tri (a2 + b2) + a2 := by rfl\n _ < tri (a2 + b2) + (a2 + b2) + 1 := by linarith\n _ = tri (a2 + b2 + 1) := (tri_step _).symm\n _ \u2264 tri (a1 + b1) := tri_incr h3\n _ \u2264 tri (a1 + b1) + a1 := by linarith\n _ = fnnn (a1, b1) := by rfl\n linarith\n done", "proofType": "tactic", "proofLengthLines": 12, "proofLengthTokens": 415}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap8Part1\nnamespace HTPI\n\n/- Definitions and theorems in Chap8Part1 and HTPIDefs\n\ntheorem zero_elts_iff_empty {A : Type} (X : Set A) :\n numElts X 0 \u2194 empty X\n\ntheorem one_elt_iff_singleton {A : Type} (X : Set A) :\n numElts X 1 \u2194 \u2203 (x : A), X = {x}\n\ntheorem nonempty_of_pos_numElts {A : Type} {X : Set A} {n : Nat}\n (h1 : numElts X n) (h2 : n > 0) : \u2203 (x : A), x \u2208 X\n\ntheorem remove_one_numElts {A : Type} {X : Set A} {n : Nat} {x : A}\n (h1 : numElts X (n + 1)) (h2 : x \u2208 X) : numElts (X \\ {x}) n\n\ndef sum_seq {A : Type} [AddZeroClass A] (m k : Nat) (f : Nat \u2192 A) : A :=\n match m with\n | 0 => 0\n | n + 1 => sum_seq n k f + f (k + n)\n\ntheorem sum_base {A : Type} [AddZeroClass A] {k : Nat} {f : Nat \u2192 A} :\n Sum i from k to k, f i = f k\n\ntheorem sum_step {A : Type} [AddZeroClass A] {k n : Nat} {f : Nat \u2192 A}\n (h : k \u2264 n) : Sum i from k to (n + 1), f i = (Sum i from k to n, f i) + f (n + 1)\n\ntheorem sum_from_zero_step {A : Type} [AddZeroClass A] {n : Nat} {f : Nat \u2192 A} :\n Sum i from 0 to (n + 1), f i = (Sum i from 0 to n, f i) + f (n + 1)\n-/\n\n/- Definitions -/\ndef nat_even (n : Nat) : Prop := \u2203 (k : Nat), n = 2 * k\n\ndef nat_odd (n : Nat) : Prop := \u2203 (k : Nat), n = 2 * k + 1\n\ndef extendPO {A : Type} (R : BinRel A) (b : A) (x y : A) : Prop :=\n R x y \u2228 (R x b \u2227 \u00acR y b)\n\ndef fact (k : Nat) : Nat :=\n match k with\n | 0 => 1\n | n + 1 => (n + 1) * fact n\n\ndef Fib (n : Nat) : Nat :=\n match n with\n | 0 => 0\n | 1 => 1\n | k + 2 => Fib k + Fib (k + 1)\n\ndef rep_image {A : Type} (f : A \u2192 A) (n : Nat) (B : Set A) : Set A :=\n match n with\n | 0 => B\n | k + 1 => image f (rep_image f k B)\n\ndef cumul_image {A : Type} (f : A \u2192 A) (B : Set A) : Set A :=\n {x : A | \u2203 (n : Nat), x \u2208 rep_image f n B}\n\n/- Section 6.1 -/\ntheorem Like_Example_6_1_2 :\n \u2200 (n : Nat), 3 \u2223 n ^ 3 + 2 * n := by\n by_induc\n \u00b7 -- Base Case\n define --Goal : \u2203 (c : Nat), 0 ^ 3 + 2 * 0 = 3 * c\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : 3 \u2223 n ^ 3 + 2 * n\n define at ih --ih : \u2203 (c : Nat), n ^ 3 + 2 * n = 3 * c\n obtain (k : Nat) (h1 : n ^ 3 + 2 * n = 3 * k) from ih\n define --Goal : \u2203 (c : Nat), (n + 1) ^ 3 + 2 * (n + 1) = 3 * c\n apply Exists.intro (k + n ^ 2 + n + 1)\n show (n + 1) ^ 3 + 2 * (n + 1) = 3 * (k + n ^ 2 + n + 1) from\n calc (n + 1) ^ 3 + 2 * (n + 1)\n _ = n ^ 3 + 2 * n + 3 * n ^ 2 + 3 * n + 3 := by ring\n _ = 3 * k + 3 * n ^ 2 + 3 * n + 3 := by rw [h1]\n _ = 3 * (k + n ^ 2 + n + 1) := by ring\n done\n done\n\ntheorem Like_Example_6_1_1 :\n \u2200 (n : Nat), (Sum i from 0 to n, 2 ^ i) + 1 = 2 ^ (n + 1) := by\n by_induc\n \u00b7 -- Base Case\n rewrite [sum_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (Sum i from 0 to n, 2 ^ i) + 1 = 2 ^ (n + 1)\n show (Sum i from 0 to n + 1, 2 ^ i) + 1 = 2 ^ (n + 1 + 1) from\n calc (Sum i from 0 to n + 1, 2 ^ i) + 1\n _ = (Sum i from 0 to n, 2 ^ i) + 2 ^ (n + 1) + 1 := by\n rw [sum_from_zero_step]\n _ = (Sum i from 0 to n, 2 ^ i) + 1 + 2 ^ (n + 1) := by ring\n _ = 2 ^ (n + 1) + 2 ^ (n + 1) := by rw [ih]\n _ = 2 ^ (n + 1 + 1) := by ring\n done\n done\n\ntheorem Example_6_1_3 : \u2200 n \u2265 5, 2 ^ n > n ^ 2 := by\n by_induc\n \u00b7 -- Base Case\n decide\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume h1 : n \u2265 5\n assume ih : 2 ^ n > n ^ 2\n have h2 : n * n \u2265 5 * n := Nat.mul_le_mul_right n h1\n show 2 ^ (n + 1) > (n + 1) ^ 2 from\n calc 2 ^ (n + 1)\n _ = 2 * 2 ^ n := by ring\n _ > 2 * n ^ 2 := by linarith\n _ \u2265 n ^ 2 + 5 * n := by linarith\n _ > n ^ 2 + 2 * n + 1 := by linarith\n _ = (n + 1) ^ 2 := by ring\n done\n done\n\n#eval 2 - 3 --Answer: 0\n\ntheorem Example_6_1_1 :\n \u2200 (n : Nat), Sum i from 0 to n, (2 : Int) ^ i =\n (2 : Int) ^ (n + 1) - (1 : Int) := by\n by_induc\n \u00b7 -- Base Case\n rewrite [sum_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : Sum i from 0 to n, (2 : Int) ^ i =\n (2 : Int) ^ (n + 1) - (1 : Int)\n show Sum i from 0 to n + 1, (2 : Int) ^ i =\n (2 : Int) ^ (n + 1 + 1) - (1 : Int) from\n calc Sum i from 0 to n + 1, (2 : Int) ^ i\n _ = (Sum i from 0 to n, (2 : Int) ^ i)\n + (2 : Int) ^ (n + 1) := by rw [sum_from_zero_step]\n _ = (2 : Int) ^ (n + 1) - (1 : Int)\n + (2 : Int) ^ (n + 1) := by rw [ih]\n _ = (2 : Int) ^ (n + 1 + 1) - (1 : Int) := by ring\n done\n done\n\n/- Section 6.2 -/\nlemma Lemma_6_2_1_1 {A : Type} {R : BinRel A} {B : Set A} {b c : A}\n (h1 : partial_order R) (h2 : b \u2208 B) (h3 : minimalElt R c (B \\ {b}))\n (h4 : R b c) : minimalElt R b B := by\n define at h3\n --h3 : c \u2208 B \\ {b} \u2227 \u00ac\u2203 x \u2208 B \\ {b}, R x c \u2227 x \u2260 c\n define --Goal : b \u2208 B \u2227 \u00ac\u2203 x \u2208 B, R x b \u2227 x \u2260 b\n apply And.intro h2 --Goal : \u00ac\u2203 x \u2208 B, R x b \u2227 x \u2260 b\n contradict h3.right with h5\n obtain (x : A) (h6 : x \u2208 B \u2227 R x b \u2227 x \u2260 b) from h5\n apply Exists.intro x --Goal : x \u2208 B \\ {b} \u2227 R x c \u2227 x \u2260 c\n apply And.intro\n \u00b7 -- Proof that x \u2208 B \\ {b}\n show x \u2208 B \\ {b} from And.intro h6.left h6.right.right\n done\n \u00b7 -- Proof that R x c \u2227 x \u2260 c\n have Rtrans : transitive R := h1.right.left\n have h7 : R x c := Rtrans x b c h6.right.left h4\n apply And.intro h7\n by_contra h8\n rewrite [h8] at h6 --h6 : c \u2208 B \u2227 R c b \u2227 c \u2260 b\n have Rantisymm : antisymmetric R := h1.right.right\n have h9 : c = b := Rantisymm c b h6.right.left h4\n show False from h6.right.right h9\n done\n done\n\nlemma Lemma_6_2_1_2 {A : Type} {R : BinRel A} {B : Set A} {b c : A}\n (h1 : partial_order R) (h2 : b \u2208 B) (h3 : minimalElt R c (B \\ {b}))\n (h4 : \u00acR b c) : minimalElt R c B := sorry\n\ntheorem Example_6_2_1 {A : Type} (R : BinRel A) (h : partial_order R) :\n \u2200 n \u2265 1, \u2200 (B : Set A), numElts B n \u2192\n \u2203 (x : A), minimalElt R x B := by\n by_induc\n \u00b7 -- Base Case\n fix B : Set A\n assume h2 : numElts B 1\n rewrite [one_elt_iff_singleton] at h2\n obtain (b : A) (h3 : B = {b}) from h2\n apply Exists.intro b\n define --Goal : b \u2208 B \u2227 \u00ac\u2203 x \u2208 B, R x b \u2227 x \u2260 b\n apply And.intro\n \u00b7 -- Proof that b \u2208 B\n rewrite [h3] --Goal : b \u2208 {b}\n define --Goal : b = b\n rfl\n done\n \u00b7 -- Proof that nothing in B is smaller than b\n by_contra h4\n obtain (x : A) (h5 : x \u2208 B \u2227 R x b \u2227 x \u2260 b) from h4\n have h6 : x \u2208 B := h5.left\n rewrite [h3] at h6 --h6 : x \u2208 {b}\n define at h6 --h6 : x = b\n show False from h5.right.right h6\n done\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume h2 : n \u2265 1\n assume ih : \u2200 (B : Set A), numElts B n \u2192 \u2203 (x : A), minimalElt R x B\n fix B : Set A\n assume h3 : numElts B (n + 1)\n have h4 : n + 1 > 0 := by linarith\n obtain (b : A) (h5 : b \u2208 B) from nonempty_of_pos_numElts h3 h4\n set B' : Set A := B \\ {b}\n have h6 : numElts B' n := remove_one_numElts h3 h5\n obtain (c : A) (h7 : minimalElt R c B') from ih B' h6\n by_cases h8 : R b c\n \u00b7 -- Case 1. h8 : R b c\n have h9 : minimalElt R b B := Lemma_6_2_1_1 h h5 h7 h8\n show \u2203 (x : A), minimalElt R x B from Exists.intro b h9\n done\n \u00b7 -- Case 2. h8 : \u00acR b c\n have h9 : minimalElt R c B := Lemma_6_2_1_2 h h5 h7 h8\n show \u2203 (x : A), minimalElt R x B from Exists.intro c h9\n done\n done\n done\n\nlemma extendPO_is_ref {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : reflexive (extendPO R b) := sorry\n\nlemma extendPO_is_trans {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : transitive (extendPO R b) := sorry\n\nlemma extendPO_is_antisymm {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : antisymmetric (extendPO R b) := sorry\n\nlemma extendPO_is_po {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : partial_order (extendPO R b) :=\n And.intro (extendPO_is_ref R b h)\n (And.intro (extendPO_is_trans R b h) (extendPO_is_antisymm R b h))\n\nlemma extendPO_extends {A : Type} (R : BinRel A) (b : A) (x y : A) :\n R x y \u2192 extendPO R b x y := by\n assume h1 : R x y\n define\n show R x y \u2228 R x b \u2227 \u00acR y b from Or.inl h1\n done\n\nlemma extendPO_all_comp {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) :\n \u2200 (x : A), extendPO R b b x \u2228 extendPO R b x b := by\n have Rref : reflexive R := h.left\n fix x : A\n or_left with h1\n define at h1 --h1 : \u00ac(R x b \u2228 R x b \u2227 \u00acR b b)\n demorgan at h1 --h1 : \u00acR x b \u2227 \u00ac(R x b \u2227 \u00acR b b)\n define --Goal : R b x \u2228 R b b \u2227 \u00acR x b\n apply Or.inr\n show R b b \u2227 \u00acR x b from And.intro (Rref b) h1.left\n done\n\ntheorem Exercise_6_2_2 {A : Type} (R : BinRel A) (h : partial_order R) :\n \u2200 (n : Nat) (B : Set A), numElts B n \u2192 \u2203 (T : BinRel A),\n partial_order T \u2227 (\u2200 (x y : A), R x y \u2192 T x y) \u2227\n \u2200 x \u2208 B, \u2200 (y : A), T x y \u2228 T y x := by\n by_induc\n \u00b7 -- Base Case\n fix B : Set A\n assume h2 : numElts B 0\n rewrite [zero_elts_iff_empty] at h2\n define at h2 --h2 : \u00ac\u2203 (x : A), x \u2208 B\n apply Exists.intro R\n apply And.intro h\n apply And.intro\n \u00b7 -- Proof that R extends R\n fix x : A; fix y : A\n assume h3 : R x y\n show R x y from h3\n done\n \u00b7 -- Proof that everything in B comparable to everything under R\n fix x : A\n assume h3 : x \u2208 B\n contradict h2\n show \u2203 (x : A), x \u2208 B from Exists.intro x h3\n done\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (B : Set A), numElts B n \u2192 \u2203 (T : BinRel A),\n partial_order T \u2227 (\u2200 (x y : A), R x y \u2192 T x y) \u2227\n \u2200 (x : A), x \u2208 B \u2192 \u2200 (y : A), T x y \u2228 T y x\n fix B : Set A\n assume h2 : numElts B (n + 1)\n have h3 : n + 1 > 0 := by linarith\n obtain (b : A) (h4 : b \u2208 B) from nonempty_of_pos_numElts h2 h3\n set B' : Set A := B \\ {b}\n have h5 : numElts B' n := remove_one_numElts h2 h4\n have h6 : \u2203 (T : BinRel A), partial_order T \u2227\n (\u2200 (x y : A), R x y \u2192 T x y) \u2227\n \u2200 (x : A), x \u2208 B' \u2192 \u2200 (y : A), T x y \u2228 T y x := ih B' h5\n obtain (T' : BinRel A)\n (h7 : partial_order T' \u2227 (\u2200 (x y : A), R x y \u2192 T' x y) \u2227\n \u2200 (x : A), x \u2208 B' \u2192 \u2200 (y : A), T' x y \u2228 T' y x) from h6\n have T'po : partial_order T' := h7.left\n have T'extR : \u2200 (x y : A), R x y \u2192 T' x y := h7.right.left\n have T'compB' : \u2200 (x : A), x \u2208 B' \u2192\n \u2200 (y : A), T' x y \u2228 T' y x := h7.right.right\n set T : BinRel A := extendPO T' b\n apply Exists.intro T\n apply And.intro (extendPO_is_po T' b T'po)\n apply And.intro\n \u00b7 -- Proof that T extends R\n fix x : A; fix y : A\n assume h8 : R x y\n have h9 : T' x y := T'extR x y h8\n show T x y from (extendPO_extends T' b x y h9)\n done\n \u00b7 -- Proof that everything in B comparable to everything under T\n fix x : A\n assume h8 : x \u2208 B\n by_cases h9 : x = b\n \u00b7 -- Case 1. h9 : x = b\n rewrite [h9]\n show \u2200 (y : A), T b y \u2228 T y b from extendPO_all_comp T' b T'po\n done\n \u00b7 -- Case 2. h9 : x \u2260 b\n have h10 : x \u2208 B' := And.intro h8 h9\n fix y : A\n have h11 : T' x y \u2228 T' y x := T'compB' x h10 y\n by_cases on h11\n \u00b7 -- Case 2.1. h11 : T' x y\n show T x y \u2228 T y x from\n Or.inl (extendPO_extends T' b x y h11)\n done\n \u00b7 -- Case 2.2. h11 : T' y x\n show T x y \u2228 T y x from\n Or.inr (extendPO_extends T' b y x h11)\n done\n done\n done\n done\n done\n\n/- Section 6.3 -/\n#eval fact 4 --Answer: 24\n\n", "theoremStatement": "theorem Example_6_3_1 : \u2200 n \u2265 4, fact n > 2 ^ n ", "theoremName": "HTPI.Example_6_3_1", "fileCreated": {"commit": 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"Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "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.Meta.Tactic.Delta", "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.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "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.Meta.Tactic.Symm", "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.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "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", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n by_induc\n \u00b7 -- Base Case\n decide\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume h1 : n \u2265 4\n assume ih : fact n > 2 ^ n\n have h2 : n + 1 > 2 := by linarith\n show fact (n + 1) > 2 ^ (n + 1) from\n calc fact (n + 1)\n _ = (n + 1) * fact n := by rfl\n _ > (n + 1) * 2 ^ n := by rel [ih]\n _ > 2 * 2 ^ n := by rel [h2]\n _ = 2 ^ (n + 1) := by ring\n done\n done", "proofType": "tactic", "proofLengthLines": 17, "proofLengthTokens": 418}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\n", "theoremStatement": "theorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) ", "theoremName": "HTPI.dvd_mod_of_dvd_a_b", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 117, "tokenPositionInFile": 3037, "theoremPositionInFile": 27}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 39, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", 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"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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done", "proofType": "tactic", "proofLengthLines": 13, "proofLengthTokens": 538}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap4\nnamespace HTPI\n\n/- Definitions -/\ndef graph {A B : Type} (f : A \u2192 B) : Set (A \u00d7 B) :=\n {(a, b) : A \u00d7 B | f a = b}\n\ndef is_func_graph {A B : Type} (G : Set (A \u00d7 B)) : Prop :=\n \u2200 (x : A), \u2203! (y : B), (x, y) \u2208 G\n\ndef onto {A B : Type} (f : A \u2192 B) : Prop :=\n \u2200 (y : B), \u2203 (x : A), f x = y\n\ndef one_to_one {A B : Type} (f : A \u2192 B) : Prop :=\n \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n\ndef closed {A : Type} (f : A \u2192 A) (C : Set A) : Prop := \u2200 x \u2208 C, f x \u2208 C\n\ndef closure {A : Type} (f : A \u2192 A) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed f D}\n\ndef closed2 {A : Type} (f : A \u2192 A \u2192 A) (C : Set A) : Prop :=\n \u2200 x \u2208 C, \u2200 y \u2208 C, f x y \u2208 C\n\ndef closure2 {A : Type} (f : A \u2192 A \u2192 A) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed2 f D}\n\ndef closed_family {A : Type} (F : Set (A \u2192 A)) (C : Set A) : Prop :=\n \u2200 f \u2208 F, closed f C\n\ndef closure_family {A : Type} (F : Set (A \u2192 A)) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed_family F D}\n\ndef image {A B : Type} (f : A \u2192 B) (X : Set A) : Set B :=\n {f x | x \u2208 X}\n\ndef inverse_image {A B : Type} (f : A \u2192 B) (Y : Set B) : Set A :=\n {a : A | f a \u2208 Y}\n\n/- Section 5.1 -/\ntheorem graph_def {A B : Type} (f : A \u2192 B) (a : A) (b : B) :\n (a, b) \u2208 graph f \u2194 f a = b := by rfl\n\ntheorem func_from_graph_ltr {A B : Type} (F : Set (A \u00d7 B)) :\n (\u2203 (f : A \u2192 B), graph f = F) \u2192 is_func_graph F := sorry\n\n--This proof is explained in Section 8.2\ntheorem func_from_graph_rtl {A B : Type} (F : Set (A \u00d7 B)) :\n is_func_graph F \u2192 (\u2203 (f : A \u2192 B), graph f = F) := by\n assume h1 : is_func_graph F\n define at h1 --h1 : \u2200 (x : A), \u2203! (y : B), (x, y) \u2208 F\n have h2 : \u2200 (x : A), \u2203 (y : B), (x, y) \u2208 F := by\n fix x : A\n obtain (y : B) (h3 : (x, y) \u2208 F)\n (h4 : \u2200 (y1 y2 : B), (x, y1) \u2208 F \u2192 (x, y2) \u2208 F \u2192 y1 = y2) from h1 x\n show \u2203 (y : B), (x, y) \u2208 F from Exists.intro y h3\n done\n set f : A \u2192 B := fun (x : A) => Classical.choose (h2 x)\n apply Exists.intro f\n apply Set.ext\n fix (x, y) : A \u00d7 B\n have h3 : (x, f x) \u2208 F := Classical.choose_spec (h2 x)\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h4 : (x, y) \u2208 graph f\n define at h4 --h4 : f x = y\n rewrite [h4] at h3\n show (x, y) \u2208 F from h3\n done\n \u00b7 -- (\u2190)\n assume h4 : (x, y) \u2208 F\n define --Goal : f x = y\n obtain (z : B) (h5 : (x, z) \u2208 F)\n (h6 : \u2200 (y1 y2 : B), (x, y1) \u2208 F \u2192 (x, y2) \u2208 F \u2192 y1 = y2) from h1 x\n show f x = y from h6 (f x) y h3 h4\n done\n done\n\ntheorem func_from_graph {A B : Type} (F : Set (A \u00d7 B)) :\n (\u2203 (f : A \u2192 B), graph f = F) \u2194 is_func_graph F :=\n Iff.intro (func_from_graph_ltr F) (func_from_graph_rtl F)\n\ntheorem Theorem_5_1_4 {A B : Type} (f g : A \u2192 B) :\n (\u2200 (a : A), f a = g a) \u2192 f = g := funext\n\nexample {A B : Type} (f g : A \u2192 B) :\n graph f = graph g \u2192 f = g := by\n assume h1 : graph f = graph g --Goal : f = g\n apply funext --Goal : \u2200 (x : A), f x = g x\n fix x : A\n have h2 : (x, f x) \u2208 graph f := by\n define --Goal : f x = f x\n rfl\n done\n rewrite [h1] at h2 --h2 : (x, f x) \u2208 graph g\n define at h2 --h2 : g x = f x\n show f x = g x from h2.symm\n done\n\ndef square1 (n : Nat) : Nat := n ^ 2\n\ndef square2 : Nat \u2192 Nat := fun (n : Nat) => n ^ 2\n\nexample : square1 = square2 := by rfl\n\n#eval square1 7 --Answer: 49\n\ntheorem Theorem_5_1_5 {A B C : Type} (f : A \u2192 B) (g : B \u2192 C) :\n \u2203 (h : A \u2192 C), graph h = comp (graph g) (graph f) := by\n set h : A \u2192 C := fun (x : A) => g (f x)\n apply Exists.intro h\n apply Set.ext\n fix (a, c) : A \u00d7 C\n apply Iff.intro\n \u00b7 -- Proof that (a, c) \u2208 graph h \u2192 (a, c) \u2208 comp (graph g) (graph f)\n assume h1 : (a, c) \u2208 graph h\n define at h1 --h1 : h a = c\n define --Goal : \u2203 (x : B), (a, x) \u2208 graph f \u2227 (x, c) \u2208 graph g\n apply Exists.intro (f a)\n apply And.intro\n \u00b7 -- Proof that (a, f a) \u2208 graph f\n define\n rfl\n done\n \u00b7 -- Proof that (f a, c) \u2208 graph g\n define\n show g (f a) = c from h1\n done\n done\n \u00b7 -- Proof that (a, c) \u2208 comp (graph g) (graph f) \u2192 (a, c) \u2208 graph h\n assume h1 : (a, c) \u2208 comp (graph g) (graph f)\n define --Goal : h a = c\n define at h1 --h1 : \u2203 (x : B), (a, x) \u2208 graph f \u2227 (x, c) \u2208 graph g\n obtain (b : B) (h2 : (a, b) \u2208 graph f \u2227 (b, c) \u2208 graph g) from h1\n have h3 : (a, b) \u2208 graph f := h2.left\n have h4 : (b, c) \u2208 graph g := h2.right\n define at h3 --h3 : f a = b\n define at h4 --h4 : g b = c\n rewrite [\u2190h3] at h4 --h4 : g (f a) = c\n show h a = c from h4\n done\n done\n\nexample {A B C D : Type} (f : A \u2192 B) (g : B \u2192 C) (h : C \u2192 D) :\n h \u2218 (g \u2218 f) = (h \u2218 g) \u2218 f := by rfl\n\nexample {A B : Type} (f : A \u2192 B) : f \u2218 id = f := by rfl\n\nexample {A B : Type} (f : A \u2192 B) : id \u2218 f = f := by rfl\n\n/- Section 5.2 -/\ntheorem Theorem_5_2_5_1 {A B C : Type} (f : A \u2192 B) (g : B \u2192 C) :\n one_to_one f \u2192 one_to_one g \u2192 one_to_one (g \u2218 f) := by\n assume h1 : one_to_one f\n assume h2 : one_to_one g\n define at h1 --h1 : \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n define at h2 --h2 : \u2200 (x1 x2 : B), g x1 = g x2 \u2192 x1 = x2\n define --Goal : \u2200 (x1 x2 : A), (g \u2218 f) x1 = (g \u2218 f) x2 \u2192 x1 = x2\n fix a1 : A\n fix a2 : A --Goal : (g \u2218 f) a1 = (g \u2218 f) a2 \u2192 a1 = a2\n define : (g \u2218 f) a1; define : (g \u2218 f) a2\n --Goal : g (f a1) = g (f a2) \u2192 a1 = a2\n assume h3 : g (f a1) = g (f a2)\n have h4 : f a1 = f a2 := h2 (f a1) (f a2) h3\n show a1 = a2 from h1 a1 a2 h4\n done\n\nlemma comp_def {A B C : Type} (g : B \u2192 C) (f : A \u2192 B) (x : A) :\n (g \u2218 f) x = g (f x) := by rfl\n\ntheorem Theorem_5_2_5_2 {A B C : Type} (f : A \u2192 B) (g : B \u2192 C) :\n onto f \u2192 onto g \u2192 onto (g \u2218 f) := by\n assume h1 : onto f\n assume h2 : onto g\n define at h1 --h1 : \u2200 (y : B), \u2203 (x : A), f x = y\n define at h2 --h2 : \u2200 (y : C), \u2203 (x : B), g x = y\n define --Goal : \u2200 (y : C), \u2203 (x : A), (g \u2218 f) x = y\n fix c : C\n obtain (b : B) (h3 : g b = c) from h2 c\n obtain (a : A) (h4 : f a = b) from h1 b\n apply Exists.intro a --Goal : (g \u2218 f) a = c\n rewrite [comp_def] --Goal : g (f a) = c\n rewrite [\u2190h4] at h3\n show g (f a) = c from h3\n done\n\n/- Section 5.3 -/\ntheorem Theorem_5_3_1 {A B : Type}\n (f : A \u2192 B) (h1 : one_to_one f) (h2 : onto f) :\n \u2203 (g : B \u2192 A), graph g = inv (graph f) := by\n rewrite [func_from_graph] --Goal : is_func_graph (inv (graph f))\n define --Goal : \u2200 (x : B), \u2203! (y : A), (x, y) \u2208 inv (graph f)\n fix b : B\n exists_unique\n \u00b7 -- Existence\n define at h2 --h2 : \u2200 (y : B), \u2203 (x : A), f x = y\n obtain (a : A) (h4 : f a = b) from h2 b\n apply Exists.intro a --Goal : (b, a) \u2208 inv (graph f)\n define --Goal : f a = b\n show f a = b from h4\n done\n \u00b7 -- Uniqueness\n fix a1 : A; fix a2 : A\n assume h3 : (b, a1) \u2208 inv (graph f)\n assume h4 : (b, a2) \u2208 inv (graph f) --Goal : a1 = a2\n define at h3 --h3 : f a1 = b\n define at h4 --h4 : f a2 = b\n rewrite [\u2190h4] at h3 --h3 : f a1 = f a2\n define at h1 --h1 : \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n show a1 = a2 from h1 a1 a2 h3\n done\n done\n\ntheorem Theorem_5_3_2_1 {A B : Type} (f : A \u2192 B) (g : B \u2192 A)\n (h1 : graph g = inv (graph f)) : g \u2218 f = id := by\n apply funext --Goal : \u2200 (x : A), (g \u2218 f) x = id x\n fix a : A --Goal : (g \u2218 f) a = id a\n have h2 : (f a, a) \u2208 graph g := by\n rewrite [h1] --Goal : (f a, a) \u2208 inv (graph f)\n define --Goal : f a = f a\n rfl\n done\n define at h2 --h2 : g (f a) = a\n show (g \u2218 f) a = id a from h2\n done\n\ntheorem Theorem_5_3_2_2 {A B : Type} (f : A \u2192 B) (g : B \u2192 A)\n (h1 : graph g = inv (graph f)) : f \u2218 g = id := sorry\n\ntheorem Theorem_5_3_3_1 {A B : Type} (f : A \u2192 B) (g : B \u2192 A)\n (h1 : g \u2218 f = id) : one_to_one f := by\n define --Goal : \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n fix a1 : A; fix a2 : A\n assume h2 : f a1 = f a2\n show a1 = a2 from\n calc a1\n _ = id a1 := by rfl\n _ = (g \u2218 f) a1 := by rw [h1]\n _ = g (f a1) := by rfl\n _ = g (f a2) := by rw [h2]\n _ = (g \u2218 f) a2 := by rfl\n _ = id a2 := by rw [h1]\n _ = a2 := by rfl\n done\n\ntheorem Theorem_5_3_3_2 {A B : Type} (f : A \u2192 B) (g : B \u2192 A)\n (h1 : f \u2218 g = id) : onto f := sorry\n\ntheorem Theorem_5_3_5 {A B : Type} (f : A \u2192 B) (g : B \u2192 A)\n (h1 : g \u2218 f = id) (h2 : f \u2218 g = id) : graph g = inv (graph f) := by\n have h3 : one_to_one f := Theorem_5_3_3_1 f g h1\n have h4 : onto f := Theorem_5_3_3_2 f g h2\n obtain (g' : B \u2192 A) (h5 : graph g' = inv (graph f))\n from Theorem_5_3_1 f h3 h4\n have h6 : g' \u2218 f = id := Theorem_5_3_2_1 f g' h5\n have h7 : g = g' :=\n calc g\n _ = id \u2218 g := by rfl\n _ = (g' \u2218 f) \u2218 g := by rw [h6]\n _ = g' \u2218 (f \u2218 g) := by rfl\n _ = g' \u2218 id := by rw [h2]\n _ = g' := by rfl\n rewrite [\u2190h7] at h5\n show graph g = inv (graph f) from h5\n done\n\n/- Section 5.4 -/\ntheorem Theorem_5_4_5 {A : Type} (f : A \u2192 A) (B : Set A) :\n \u2203 (C : Set A), closure f B C := by\n set F : Set (Set A) := {D : Set A | B \u2286 D \u2227 closed f D}\n set C : Set A := \u22c2\u2080 F\n apply Exists.intro C --Goal : closure f B C\n define --Goal : C \u2208 F \u2227 \u2200 x \u2208 F, C \u2286 x\n apply And.intro\n \u00b7 -- Proof that C \u2208 F\n define --Goal : B \u2286 C \u2227 closed f C\n apply And.intro\n \u00b7 -- Proof that B \u2286 C\n fix a : A\n assume h1 : a \u2208 B --Goal : a \u2208 C\n define --Goal : \u2200 t \u2208 F, a \u2208 t\n fix D : Set A\n assume h2 : D \u2208 F\n define at h2 --h2 : B \u2286 D \u2227 closed f D\n show a \u2208 D from h2.left h1\n done\n \u00b7 -- Proof that C is closed under f\n define --Goal : \u2200 x \u2208 C, f x \u2208 C\n fix a : A\n assume h1 : a \u2208 C --Goal : f a \u2208 C\n define --Goal : \u2200 t \u2208 F, f a \u2208 t\n fix D : Set A\n assume h2 : D \u2208 F --Goal : f a \u2208 D\n define at h1 --h1 : \u2200 t \u2208 F, a \u2208 t\n have h3 : a \u2208 D := h1 D h2\n define at h2 --h2 : B \u2286 D \u2227 closed f D\n have h4 : closed f D := h2.right\n define at h4 --h4 : \u2200 x \u2208 D, f x \u2208 D\n show f a \u2208 D from h4 a h3\n done\n done\n \u00b7 -- Proof that C is smallest\n fix D : Set A\n assume h1 : D \u2208 F --Goal : sub A C D\n define\n fix a : A\n assume h2 : a \u2208 C --Goal : a \u2208 D\n define at h2 --h2 : \u2200 t \u2208 F, a \u2208 t\n show a \u2208 D from h2 D h1\n done\n done\n\ndef plus (m n : Int) : Int := m + n\n\ndef plus' : Int \u2192 Int \u2192 Int := fun (m n : Int) => m + n\n\ndef plus'' : Int \u2192 Int \u2192 Int := fun (m : Int) => (fun (n : Int) => m + n)\n\nexample : plus = plus'' := by rfl\n\nexample : plus' = plus'' := by rfl\n\n#eval plus 3 2 --Answer: 5\n\ntheorem Theorem_5_4_9 {A : Type} (f : A \u2192 A \u2192 A) (B : Set A) :\n \u2203 (C : Set A), closure2 f B C := sorry\n\n/- Section 5.5 -/\ntheorem image_def {A B : Type} (f : A \u2192 B) (X : Set A) (b : B) :\n b \u2208 image f X \u2194 \u2203 x \u2208 X, f x = b := by rfl\n\ntheorem inverse_image_def {A B : Type} (f : A \u2192 B) (Y : Set B) (a : A) :\n a \u2208 inverse_image f Y \u2194 f a \u2208 Y := by rfl\n\ntheorem Theorem_5_5_2_1 {A B : Type} (f : A \u2192 B) (W X : Set A) :\n image f (W \u2229 X) \u2286 image f W \u2229 image f X := by\n fix y : B\n assume h1 : y \u2208 image f (W \u2229 X) --Goal : y \u2208 image f W \u2229 image f X\n define at h1 --h1 : \u2203 x \u2208 W \u2229 X, f x = y\n obtain (x : A) (h2 : x \u2208 W \u2229 X \u2227 f x = y) from h1\n define : x \u2208 W \u2229 X at h2 --h2 : (x \u2208 W \u2227 x \u2208 X) \u2227 f x = y\n apply And.intro\n \u00b7 -- Proof that y \u2208 image f W\n define --Goal : \u2203 x \u2208 W, f x = y\n show \u2203 (x : A), x \u2208 W \u2227 f x = y from\n Exists.intro x (And.intro h2.left.left h2.right)\n done\n \u00b7 -- Proof that y \u2208 image f X\n show y \u2208 image f X from\n Exists.intro x (And.intro h2.left.right h2.right)\n done\n done\n\n", "theoremStatement": "theorem Theorem_5_5_2_2 {A B : Type} (f : A \u2192 B) (W X : Set A)\n (h1 : one_to_one f) : image f (W \u2229 X) = image f W \u2229 image f X ", "theoremName": "HTPI.Theorem_5_5_2_2", "fileCreated": {"commit": "cc48e6ba4c656bf74f7690323b2c27f2c466665b", "date": "2023-03-11"}, "theoremCreated": {"commit": "190e34021c4943fc63cee05edf567875671378bb", "date": "2023-02-18"}, "file": "htpi/HTPILib/Chap5.lean", 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"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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply Set.ext\n fix y : B --Goal : y \u2208 image f (W \u2229 X) \u2194 y \u2208 image f W \u2229 image f X\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : y \u2208 image f (W \u2229 X)\n show y \u2208 image f W \u2229 image f X from Theorem_5_5_2_1 f W X h2\n done\n \u00b7 -- (\u2190)\n assume h2 : y \u2208 image f W \u2229 image f X --Goal : y \u2208 image f (W \u2229 X)\n define at h2 --h2 : y \u2208 image f W \u2227 y \u2208 image f X\n rewrite [image_def, image_def] at h2\n --h2 : (\u2203 x \u2208 W, f x = y) \u2227 \u2203 x \u2208 X, f x = y\n obtain (x1 : A) (h3 : x1 \u2208 W \u2227 f x1 = y) from h2.left\n obtain (x2 : A) (h4 : x2 \u2208 X \u2227 f x2 = y) from h2.right\n have h5 : f x2 = y := h4.right\n rewrite [\u2190h3.right] at h5 --h5 : f x2 = f x1\n define at h1 --h1 : \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n have h6 : x2 = x1 := h1 x2 x1 h5\n rewrite [h6] at h4 --h4 : x1 \u2208 X \u2227 f x1 = y\n show y \u2208 image f (W \u2229 X) from\n Exists.intro x1 (And.intro (And.intro h3.left h4.left) h3.right)\n done\n done", "proofType": "tactic", "proofLengthLines": 23, "proofLengthTokens": 977}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap8Part1\nnamespace HTPI\n\n/- Definitions and theorems in Chap8Part1 and HTPIDefs\n\ntheorem zero_elts_iff_empty {A : Type} (X : Set A) :\n numElts X 0 \u2194 empty X\n\ntheorem one_elt_iff_singleton {A : Type} (X : Set A) :\n numElts X 1 \u2194 \u2203 (x : A), X = {x}\n\ntheorem nonempty_of_pos_numElts {A : Type} {X : Set A} {n : Nat}\n (h1 : numElts X n) (h2 : n > 0) : \u2203 (x : A), x \u2208 X\n\ntheorem remove_one_numElts {A : Type} {X : Set A} {n : Nat} {x : A}\n (h1 : numElts X (n + 1)) (h2 : x \u2208 X) : numElts (X \\ {x}) n\n\ndef sum_seq {A : Type} [AddZeroClass A] (m k : Nat) (f : Nat \u2192 A) : A :=\n match m with\n | 0 => 0\n | n + 1 => sum_seq n k f + f (k + n)\n\ntheorem sum_base {A : Type} [AddZeroClass A] {k : Nat} {f : Nat \u2192 A} :\n Sum i from k to k, f i = f k\n\ntheorem sum_step {A : Type} [AddZeroClass A] {k n : Nat} {f : Nat \u2192 A}\n (h : k \u2264 n) : Sum i from k to (n + 1), f i = (Sum i from k to n, f i) + f (n + 1)\n\ntheorem sum_from_zero_step {A : Type} [AddZeroClass A] {n : Nat} {f : Nat \u2192 A} :\n Sum i from 0 to (n + 1), f i = (Sum i from 0 to n, f i) + f (n + 1)\n-/\n\n/- Definitions -/\ndef nat_even (n : Nat) : Prop := \u2203 (k : Nat), n = 2 * k\n\ndef nat_odd (n : Nat) : Prop := \u2203 (k : Nat), n = 2 * k + 1\n\ndef extendPO {A : Type} (R : BinRel A) (b : A) (x y : A) : Prop :=\n R x y \u2228 (R x b \u2227 \u00acR y b)\n\ndef fact (k : Nat) : Nat :=\n match k with\n | 0 => 1\n | n + 1 => (n + 1) * fact n\n\ndef Fib (n : Nat) : Nat :=\n match n with\n | 0 => 0\n | 1 => 1\n | k + 2 => Fib k + Fib (k + 1)\n\ndef rep_image {A : Type} (f : A \u2192 A) (n : Nat) (B : Set A) : Set A :=\n match n with\n | 0 => B\n | k + 1 => image f (rep_image f k B)\n\ndef cumul_image {A : Type} (f : A \u2192 A) (B : Set A) : Set A :=\n {x : A | \u2203 (n : Nat), x \u2208 rep_image f n B}\n\n/- Section 6.1 -/\ntheorem Like_Example_6_1_2 :\n \u2200 (n : Nat), 3 \u2223 n ^ 3 + 2 * n := by\n by_induc\n \u00b7 -- Base Case\n define --Goal : \u2203 (c : Nat), 0 ^ 3 + 2 * 0 = 3 * c\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : 3 \u2223 n ^ 3 + 2 * n\n define at ih --ih : \u2203 (c : Nat), n ^ 3 + 2 * n = 3 * c\n obtain (k : Nat) (h1 : n ^ 3 + 2 * n = 3 * k) from ih\n define --Goal : \u2203 (c : Nat), (n + 1) ^ 3 + 2 * (n + 1) = 3 * c\n apply Exists.intro (k + n ^ 2 + n + 1)\n show (n + 1) ^ 3 + 2 * (n + 1) = 3 * (k + n ^ 2 + n + 1) from\n calc (n + 1) ^ 3 + 2 * (n + 1)\n _ = n ^ 3 + 2 * n + 3 * n ^ 2 + 3 * n + 3 := by ring\n _ = 3 * k + 3 * n ^ 2 + 3 * n + 3 := by rw [h1]\n _ = 3 * (k + n ^ 2 + n + 1) := by ring\n done\n done\n\ntheorem Like_Example_6_1_1 :\n \u2200 (n : Nat), (Sum i from 0 to n, 2 ^ i) + 1 = 2 ^ (n + 1) := by\n by_induc\n \u00b7 -- Base Case\n rewrite [sum_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (Sum i from 0 to n, 2 ^ i) + 1 = 2 ^ (n + 1)\n show (Sum i from 0 to n + 1, 2 ^ i) + 1 = 2 ^ (n + 1 + 1) from\n calc (Sum i from 0 to n + 1, 2 ^ i) + 1\n _ = (Sum i from 0 to n, 2 ^ i) + 2 ^ (n + 1) + 1 := by\n rw [sum_from_zero_step]\n _ = (Sum i from 0 to n, 2 ^ i) + 1 + 2 ^ (n + 1) := by ring\n _ = 2 ^ (n + 1) + 2 ^ (n + 1) := by rw [ih]\n _ = 2 ^ (n + 1 + 1) := by ring\n done\n done\n\n", "theoremStatement": "theorem Example_6_1_3 : \u2200 n \u2265 5, 2 ^ n > n ^ 2 ", "theoremName": "HTPI.Example_6_1_3", "fileCreated": {"commit": "cc48e6ba4c656bf74f7690323b2c27f2c466665b", "date": "2023-03-11"}, "theoremCreated": {"commit": "190e34021c4943fc63cee05edf567875671378bb", "date": "2023-02-18"}, "file": "htpi/HTPILib/Chap6.lean", "module": "HTPILib.Chap6", "jsonFile": "HTPILib.Chap6.jsonl", "positionMetadata": {"lineInFile": 106, "tokenPositionInFile": 3268, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 215, "importedModules": ["Init.Prelude", "Init.Coe", 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"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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n by_induc\n \u00b7 -- Base Case\n decide\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume h1 : n \u2265 5\n assume ih : 2 ^ n > n ^ 2\n have h2 : n * n \u2265 5 * n := Nat.mul_le_mul_right n h1\n show 2 ^ (n + 1) > (n + 1) ^ 2 from\n calc 2 ^ (n + 1)\n _ = 2 * 2 ^ n := by ring\n _ > 2 * n ^ 2 := by linarith\n _ \u2265 n ^ 2 + 5 * n := by linarith\n _ > n ^ 2 + 2 * n + 1 := by linarith\n _ = (n + 1) ^ 2 := by ring\n done\n done", "proofType": "tactic", "proofLengthLines": 18, "proofLengthTokens": 470}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap7\nnamespace HTPI\n\n/- Definitions -/\ndef fnz (n : Nat) : Int := if 2 \u2223 n then \u2191(n / 2) else -\u2191((n + 1) / 2)\n\ndef fzn (a : Int) : Nat := if a \u2265 0 then 2 * Int.toNat a else 2 * Int.toNat (-a) - 1\n\ndef tri (k : Nat) : Nat := k * (k + 1) / 2\n\ndef fnnn (p : Nat \u00d7 Nat) : Nat := tri (p.1 + p.2) + p.1\n\ndef num_elts_below (A : Set Nat) (m s : Nat) : Prop :=\n match m with\n | 0 => s = 0\n | n + 1 => (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s)\n\ndef enum (A : Set Nat) (s n : Nat) : Prop := n \u2208 A \u2227 num_elts_below A n s\n\ndef unique_val_on_N {U : Type} (R : Rel Nat U) : Prop :=\n \u2200 \u2983n : Nat\u2984 \u2983x1 x2 : U\u2984, R n x1 \u2192 R n x2 \u2192 x1 = x2\n\ndef nat_rel_onto {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203 (n : Nat), R n x\n\ndef fcnl_onto_from_nat {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n unique_val_on_N R \u2227 nat_rel_onto R A\n\ndef fcnl_one_one_to_nat {U : Type} (R : Rel U Nat) (A : Set U) : Prop :=\n fcnl_on R A \u2227 \u2200 \u2983x1 x2 : U\u2984 \u2983n : Nat\u2984,\n (x1 \u2208 A \u2227 R x1 n) \u2192 (x2 \u2208 A \u2227 R x2 n) \u2192 x1 = x2\n\ndef least_rel_to {U : Type} (S : Rel Nat U) (x : U) (n : Nat) : Prop :=\n S n x \u2227 \u2200 (m : Nat), S m x \u2192 n \u2264 m\n\ndef restrict_to {U V : Type} (S : Rel U V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 S x y\n\ndef fqn (q : Rat) : Nat := fnnn (fzn q.num, q.den)\n\ndef Set_rp_below (m : Nat) : Set Nat := {n : Nat | rel_prime m n \u2227 n < m}\n\ndef Set_prod {U V : Type} (A : Set U) (B : Set V) : Set (U \u00d7 V) :=\n {(a, b) : U \u00d7 V | a \u2208 A \u2227 b \u2208 B}\n\nnotation:75 A:75 \" \u00d7\u209b \" B:75 => Set_prod A B\n\ndef Rel_prod {U V W X : Type} (R : Rel U V) (S : Rel W X)\n (p : U \u00d7 W) (q : V \u00d7 X) : Prop := R p.1 q.1 \u2227 S p.2 q.2\n\nnotation:75 R:75 \" \u00d7\u1d63 \" S:75 => Rel_prod R S\n\ndef qr (n a : Nat) : Nat \u00d7 Nat := (a / n, a % n)\n\ndef mod_mod (m n a : Nat) : Nat \u00d7 Nat := (a % m, a % n)\n\ndef enum_union_fam {U : Type}\n (F : Set (Set U)) (f : Set U \u2192 Rel Nat U) (R : Rel Nat (Set U))\n (n : Nat) (a : U) : Prop := \u2203 (p : Nat \u00d7 Nat), fnnn p = n \u2227\n \u2203 A \u2208 F, R p.1 A \u2227 (f A) p.2 a\n\ndef seq {U : Type} (A : Set U) : Set (List U) :=\n {l : List U | \u2200 x \u2208 l, x \u2208 A}\n\ndef seq_by_length {U : Type} (A : Set U) (n : Nat) : Set (List U) :=\n {l : List U | l \u2208 seq A \u2227 l.length = n}\n\ndef seq_cons (U : Type) (p : U \u00d7 (List U)) : List U := p.1 :: p.2\n\ndef sbl_set {U : Type} (A : Set U) : Set (Set (List U)) :=\n {S : Set (List U) | \u2203 (n : Nat), seq_by_length A n = S}\n\ndef rep_common_image\n {U V : Type} (R S : Rel U V) (X0 : Set U) (n : Nat) : Set U :=\n match n with\n | 0 => X0\n | m + 1 => {a : U | \u2203 x \u2208 rep_common_image R S X0 m,\n \u2203 (y : V), R x y \u2227 S a y}\n\ndef cum_rep_image {U V : Type} (R S : Rel U V) (X0 : Set U) : Set U :=\n {a : U | \u2203 (n : Nat), a \u2208 rep_common_image R S X0 n}\n\ndef csb_match {U V : Type} (R S : Rel U V) (X0 : Set U)\n (x : U) (y : V) : Prop := x \u2208 cum_rep_image R S X0 \u2227 R x y \u2228\n x \u2209 cum_rep_image R S X0 \u2227 S x y\n\n/- Section 8.1 -/\n#eval [fnz 0, fnz 1, fnz 2, fnz 3, fnz 4, fnz 5, fnz 6]\n --Answer: [0, -1, 1, -2, 2, -3, 3]\n\n#eval [fzn 0, fzn (-1), fzn 1, fzn (-2), fzn 2, fzn (-3), fzn 3]\n --Answer: [0, 1, 2, 3, 4, 5, 6]\n\nlemma fnz_even (k : Nat) : fnz (2 * k) = \u2191k := by\n have h1 : 2 \u2223 2 * k := by\n apply Exists.intro k\n rfl\n done\n have h2 : fnz (2 * k) = if 2 \u2223 2 * k then \u2191(2 * k / 2)\n else -\u2191((2 * k + 1) / 2) := by rfl\n rewrite [if_pos h1] at h2 --h2 : fnz (2 * k) = \u2191(2 * k / 2)\n have h3 : 0 < 2 := by linarith\n rewrite [Nat.mul_div_cancel_left k h3] at h2\n show fnz (2 * k) = \u2191k from h2\n done\n\nlemma fnz_odd (k : Nat) : fnz (2 * k + 1) = -\u2191(k + 1) := sorry\n\nlemma fzn_nat (k : Nat) : fzn \u2191k = 2 * k := by rfl\n\nlemma fzn_neg_succ_nat (k : Nat) : fzn (-\u2191(k + 1)) = 2 * k + 1 := by rfl\n\n--From exercises of Section 6.1\ntheorem Exercise_6_1_16a1 : \u2200 (n : Nat), nat_even n \u2228 nat_odd n := sorry\n\nlemma fzn_fnz : fzn \u2218 fnz = id := by\n apply funext --Goal : \u2200 (x : Nat), (fzn \u2218 fnz) x = id x\n fix n : Nat\n rewrite [comp_def] --Goal : fzn (fnz n) = id n\n have h1 : nat_even n \u2228 nat_odd n := Exercise_6_1_16a1 n\n by_cases on h1\n \u00b7 -- Case 1. h1 : nat_even n\n obtain (k : Nat) (h2 : n = 2 * k) from h1\n rewrite [h2, fnz_even, fzn_nat]\n rfl\n done\n \u00b7 -- Case 2. h1 : nat_odd n\n obtain (k : Nat) (h2 : n = 2 * k + 1) from h1\n rewrite [h2, fnz_odd, fzn_neg_succ_nat]\n rfl\n done\n done\n\nlemma fnz_fzn : fnz \u2218 fzn = id := sorry\n\nlemma fzn_one_one : one_to_one fzn := Theorem_5_3_3_1 fzn fnz fnz_fzn\n\nlemma fzn_onto : onto fzn := Theorem_5_3_3_2 fzn fnz fzn_fnz\n\nlemma fnz_one_one : one_to_one fnz := Theorem_5_3_3_1 fnz fzn fzn_fnz\n\nlemma fnz_onto : onto fnz := Theorem_5_3_3_2 fnz fzn fnz_fzn\n\nlemma fnnn_def (a b : Nat) : fnnn (a, b) = tri (a + b) + a := by rfl\n\n#eval [fnnn (0, 0), fnnn (0, 1), fnnn (1, 0), fnnn (0, 2), fnnn (1, 1)]\n --Answer: [0, 1, 2, 3, 4]\n\nlemma tri_step (k : Nat) : tri (k + 1) = tri k + k + 1 := sorry\n\nlemma tri_incr {j k : Nat} (h1 : j \u2264 k) : tri j \u2264 tri k := sorry\n\nlemma le_of_fnnn_eq {a1 b1 a2 b2 : Nat}\n (h1 : fnnn (a1, b1) = fnnn (a2, b2)) : a1 + b1 \u2264 a2 + b2 := by\n by_contra h2\n have h3 : a2 + b2 + 1 \u2264 a1 + b1 := by linarith\n have h4 : fnnn (a2, b2) < fnnn (a1, b1) :=\n calc fnnn (a2, b2)\n _ = tri (a2 + b2) + a2 := by rfl\n _ < tri (a2 + b2) + (a2 + b2) + 1 := by linarith\n _ = tri (a2 + b2 + 1) := (tri_step _).symm\n _ \u2264 tri (a1 + b1) := tri_incr h3\n _ \u2264 tri (a1 + b1) + a1 := by linarith\n _ = fnnn (a1, b1) := by rfl\n linarith\n done\n\nlemma fnnn_one_one : one_to_one fnnn := by\n fix (a1, b1) : Nat \u00d7 Nat\n fix (a2, b2) : Nat \u00d7 Nat\n assume h1 : fnnn (a1, b1) = fnnn (a2, b2) --Goal : (a1, b1) = (a2, b2)\n have h2 : a1 + b1 \u2264 a2 + b2 := le_of_fnnn_eq h1\n have h3 : a2 + b2 \u2264 a1 + b1 := le_of_fnnn_eq h1.symm\n have h4 : a1 + b1 = a2 + b2 := by linarith\n rewrite [fnnn_def, fnnn_def, h4] at h1\n --h1 : tri (a2 + b2) + a1 = tri (a2 + b2) + a2\n have h6 : a1 = a2 := Nat.add_left_cancel h1\n rewrite [h6] at h4 --h4 : a2 + b1 = a2 + b2\n have h7 : b1 = b2 := Nat.add_left_cancel h4\n rewrite [h6, h7]\n rfl\n done\n\nlemma fnnn_onto : onto fnnn := by\n define --Goal : \u2200 (y : Nat), \u2203 (x : Nat \u00d7 Nat), fnnn x = y\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro (0, 0)\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2203 (x : Nat \u00d7 Nat), fnnn x = n\n obtain ((a, b) : Nat \u00d7 Nat) (h1 : fnnn (a, b) = n) from ih\n by_cases h2 : b = 0\n \u00b7 -- Case 1. h2 : b = 0\n apply Exists.intro (0, a + 1)\n show fnnn (0, a + 1) = n + 1 from\n calc fnnn (0, a + 1)\n _ = tri (0 + (a + 1)) + 0 := by rfl\n _ = tri (a + 1) := by ring\n _ = tri a + a + 1 := tri_step a\n _ = tri (a + 0) + a + 1 := by ring\n _ = fnnn (a, b) + 1 := by rw [h2, fnnn_def]\n _ = n + 1 := by rw [h1]\n done\n \u00b7 -- Case 2. h2 : b \u2260 0\n obtain (k : Nat) (h3 : b = k + 1) from\n exists_eq_add_one_of_ne_zero h2\n apply Exists.intro (a + 1, k)\n show fnnn (a + 1, k) = n + 1 from\n calc fnnn (a + 1, k)\n _ = tri (a + 1 + k) + (a + 1) := by rfl\n _ = tri (a + (k + 1)) + a + 1 := by ring\n _ = tri (a + b) + a + 1 := by rw [h3]\n _ = fnnn (a, b) + 1 := by rfl\n _ = n + 1 := by rw [h1]\n done\n done\n done\n\nlemma one_one_on_of_one_one {U V : Type} {f : U \u2192 V}\n (h : one_to_one f) (A : Set U) : one_one_on f A := by\n define\n fix x1 : U; fix x2 : U\n assume h1 : x1 \u2208 A\n assume h2 : x2 \u2208 A\n show f x1 = f x2 \u2192 x1 = x2 from h x1 x2\n done\n\nlemma elt_Univ {U : Type} (u : U) :\n u \u2208 Univ U := by trivial\n\ntheorem equinum_Univ {U V : Type} {f : U \u2192 V}\n (h1 : one_to_one f) (h2 : onto f) : Univ U \u223c Univ V := by\n have h3 : image f (Univ U) = Univ V := by\n apply Set.ext\n fix v : V\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h3 : v \u2208 image f (Univ U)\n show v \u2208 Univ V from elt_Univ v\n done\n \u00b7 -- (\u2190)\n assume h3 : v \u2208 Univ V\n obtain (u : U) (h4 : f u = v) from h2 v\n apply Exists.intro u\n apply And.intro _ h4\n show u \u2208 Univ U from elt_Univ u\n done\n done\n show Univ U \u223c Univ V from\n equinum_image (one_one_on_of_one_one h1 (Univ U)) h3\n done\n\ntheorem Z_equinum_N : Univ Int \u223c Univ Nat :=\n equinum_Univ fzn_one_one fzn_onto\n\ntheorem NxN_equinum_N : Univ (Nat \u00d7 Nat) \u223c Univ Nat :=\n equinum_Univ fnnn_one_one fnnn_onto\n\nlemma neb_step (A : Set Nat) (n s : Nat) : num_elts_below A (n + 1) s \u2194\n (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s) := by rfl\n\nlemma neb_step_elt {A : Set Nat} {n : Nat} (h1 : n \u2208 A) (s : Nat) :\n num_elts_below A (n + 1) s \u2194 1 \u2264 s \u2227 num_elts_below A n (s - 1) := by\n rewrite [neb_step]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) \u2228\n \u00acn \u2208 A \u2227 num_elts_below A n s\n by_cases on h2\n \u00b7 -- Case 1. h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n show 1 \u2264 s \u2227 num_elts_below A n (s - 1) from h2.right\n done\n \u00b7 -- Case 2. h2 : \u00acn \u2208 A \u2227 num_elts_below A n s\n show 1 \u2264 s \u2227 num_elts_below A n (s - 1) from absurd h1 h2.left\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n apply Or.inl\n show n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) from And.intro h1 h2\n done\n done\n\nlemma neb_step_not_elt {A : Set Nat} {n : Nat} (h1 : n \u2209 A) (s : Nat) :\n num_elts_below A (n + 1) s \u2194 num_elts_below A n s := by\n rewrite [neb_step]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) \u2228\n \u00acn \u2208 A \u2227 num_elts_below A n s\n by_cases on h2\n \u00b7 -- Case 1. h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n show num_elts_below A n s from absurd h2.left h1\n done\n \u00b7 -- Case 2. h2 : \u00acn \u2208 A \u2227 num_elts_below A n s\n show num_elts_below A n s from h2.right\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : num_elts_below A n s\n apply Or.inr\n show \u00acn \u2208 A \u2227 num_elts_below A n s from And.intro h1 h2\n done\n done\n\nlemma neb_exists (A : Set Nat) :\n \u2200 (n : Nat), \u2203 (s : Nat), num_elts_below A n s := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro 0\n define\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2203 (s : Nat), num_elts_below A n s\n obtain (t : Nat) (h1 : num_elts_below A n t) from ih\n by_cases h2 : n \u2208 A\n \u00b7 -- Case 1. h2 : n \u2208 A\n apply Exists.intro (t + 1)\n rewrite [neb_step_elt h2, Nat.add_sub_cancel]\n apply And.intro _ h1\n linarith\n done\n \u00b7 -- Case 2. h2 : n \u2209 A\n apply Exists.intro t\n rewrite [neb_step_not_elt h2]\n show num_elts_below A n t from h1\n done\n done\n done\n\nlemma neb_unique (A : Set Nat) : \u2200 \u2983n : Nat\u2984, \u2200 \u2983s1 s2 : Nat\u2984,\n num_elts_below A n s1 \u2192 num_elts_below A n s2 \u2192 s1 = s2 := by\n by_induc\n \u00b7 -- Base Case\n fix s1 : Nat; fix s2 : Nat\n assume h1 : num_elts_below A 0 s1\n assume h2 : num_elts_below A 0 s2\n define at h1; define at h2 --h1 : s1 = 0; h2 : s2 = 0\n rewrite [h1, h2]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 \u2983s1 s2 : Nat\u2984,\n num_elts_below A n s1 \u2192 num_elts_below A n s2 \u2192 s1 = s2\n fix s1 : Nat; fix s2 : Nat\n assume h1 : num_elts_below A (n + 1) s1\n assume h2 : num_elts_below A (n + 1) s2\n by_cases h3 : n \u2208 A\n \u00b7 -- Case 1. h3 : n \u2208 A\n rewrite [neb_step_elt h3] at h1\n rewrite [neb_step_elt h3] at h2\n --h1 : 1 \u2264 s1 \u2227 num_elts_below A n (s1 - 1)\n --h2 : 1 \u2264 s2 \u2227 num_elts_below A n (s2 - 1)\n have h4 : s1 - 1 = s2 - 1 := ih h1.right h2.right\n show s1 = s2 from\n calc s1\n _ = s1 - 1 + 1 := (Nat.sub_add_cancel h1.left).symm\n _ = s2 - 1 + 1 := by rw [h4]\n _ = s2 := Nat.sub_add_cancel h2.left\n done\n \u00b7 -- Case 2. h3 : n \u2209 A\n rewrite [neb_step_not_elt h3] at h1 --h1 : num_elts_below A n s1\n rewrite [neb_step_not_elt h3] at h2 --h2 : num_elts_below A n s2\n show s1 = s2 from ih h1 h2\n done\n done\n done\n\nlemma neb_increase {A : Set Nat} {n s : Nat} (h1 : enum A s n) :\n \u2200 \u2983m : Nat\u2984, m \u2265 n + 1 \u2192 \u2200 \u2983t : Nat\u2984, num_elts_below A m t \u2192 s < t := by\n by_induc\n \u00b7 -- Base Case\n define at h1\n fix t : Nat\n assume h2 : num_elts_below A (n + 1) t\n rewrite [neb_step_elt h1.left] at h2\n have h3 : s = t - 1 := neb_unique A h1.right h2.right\n show s < t from\n calc s\n _ = t - 1 := h3\n _ < t - 1 + 1 := by linarith\n _ = t := Nat.sub_add_cancel h2.left\n done\n \u00b7 -- Induction Step\n fix m : Nat\n assume h2 : m \u2265 n + 1\n assume ih : \u2200 \u2983t : Nat\u2984, num_elts_below A m t \u2192 s < t\n fix t : Nat\n assume h3 : num_elts_below A (m + 1) t\n by_cases h4 : m \u2208 A\n \u00b7 -- Case 1. h4 : m \u2208 A\n rewrite [neb_step_elt h4] at h3\n have h5 : s < t - 1 := ih h3.right\n show s < t from\n calc s\n _ < t - 1 := h5\n _ \u2264 t := Nat.sub_le _ _\n done\n \u00b7 -- Case 2. h4 : m \u2209 A\n rewrite [neb_step_not_elt h4] at h3\n show s < t from ih h3\n done\n done\n done\n\nlemma enum_not_skip {A : Set Nat} : \u2200 \u2983m s : Nat\u2984, num_elts_below A m s \u2192\n \u2200 t < s, \u2203 (n : Nat), enum A t n := by\n by_induc\n \u00b7 -- Base Case\n fix s : Nat\n assume h1 : num_elts_below A 0 s\n define at h1\n fix t : Nat\n contrapos\n assume h2 : \u00ac\u2203 (n : Nat), enum A t n\n linarith\n done\n \u00b7 -- Induction Step\n fix m : Nat\n assume ih : \u2200 \u2983s : Nat\u2984, num_elts_below A m s \u2192 \u2200 t < s, \u2203 (n : Nat), enum A t n\n fix s : Nat\n assume h1 : num_elts_below A (m + 1) s\n by_cases h2 : m \u2208 A\n \u00b7 -- Case 1. h2 : m \u2208 A\n rewrite [neb_step_elt h2] at h1\n have h3 : \u2200 t < s - 1, \u2203 (n : Nat), enum A t n := ih h1.right\n fix t : Nat\n assume h4 : t < s\n by_cases h5 : t = s - 1\n \u00b7 -- Case 1.1. h5 : t = s - 1\n apply Exists.intro m\n define\n apply And.intro h2\n rewrite [h5]\n show num_elts_below A m (s - 1) from h1.right\n done\n \u00b7 -- Case 1.2. h5 : t \u2260 s - 1\n have h6 : t \u2264 s - 1 := Nat.le_pred_of_lt h4\n have h7 : t < s - 1 := Nat.lt_of_le_of_ne h6 h5\n show \u2203 (n : Nat), enum A t n from ih h1.right t h7\n done\n done\n \u00b7 -- Case 2. h2 : m \u2209 A\n rewrite [neb_step_not_elt h2] at h1\n show \u2200 t < s, \u2203 (n : Nat), enum A t n from ih h1\n done\n done\n done\n\nlemma enum_le {A : Set Nat} {t n1 n2 : Nat}\n (h1 : enum A t n1) (h2 : enum A t n2) : n1 \u2264 n2 := by\n by_contra h3\n have h4 : n2 + 1 \u2264 n1 := by linarith\n define at h1\n have h5 : t < t := neb_increase h2 h4 h1.right\n linarith\n done\n\nlemma enum_unique (A : Set Nat) (t : Nat) :\n \u2200 \u2983n1 n2 : Nat\u2984, enum A t n1 \u2192 enum A t n2 \u2192 n1 = n2 := by\n fix n1 : Nat; fix n2 : Nat\n assume h1 : enum A t n1\n assume h2 : enum A t n2\n have h3 : n1 \u2264 n2 := enum_le h1 h2\n have h4 : n2 \u2264 n1 := enum_le h2 h1\n linarith\n done\n\nlemma inv_enum_fcnl (A : Set Nat) : fcnl_on (invRel (enum A)) A := by\n define\n fix n : Nat\n assume h1 : n \u2208 A\n exists_unique\n \u00b7 -- Existence\n obtain (s : Nat) (h2 : num_elts_below A n s) from neb_exists A n\n apply Exists.intro s\n define\n show n \u2208 A \u2227 num_elts_below A n s from And.intro h1 h2\n done\n \u00b7 -- Uniqueness\n fix s1 : Nat; fix s2 : Nat\n assume h2 : invRel (enum A) n s1\n assume h3 : invRel (enum A) n s2\n define at h2; define at h3\n show s1 = s2 from neb_unique A h2.right h3.right\n done\n done\n\nlemma bdd_subset_nat_match {A : Set Nat} {m s : Nat}\n (h1 : \u2200 n \u2208 A, n < m) (h2 : num_elts_below A m s) :\n matching (enum A) (I s) A := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix t : Nat; fix n : Nat\n assume h3 : enum A t n\n have h4 : \u2200 \u2983m : Nat\u2984, m \u2265 n + 1 \u2192 \u2200 \u2983t_1 : Nat\u2984, num_elts_below A m t_1 \u2192 t < t_1 :=\n neb_increase h3\n define at h3\n apply And.intro _ h3.left\n define\n have h5 : n < m := h1 n h3.left\n have h6 : m \u2265 n + 1 := h5\n show t < s from h4 h6 h2\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- proof of fcnl_on (enum A) (I s)\n define\n fix t : Nat\n assume h3 : t \u2208 I s\n define at h3\n exists_unique\n \u00b7 -- Existence\n show \u2203 (y : Nat), enum A t y from enum_not_skip h2 t h3\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A t y_1 \u2192 enum A t y_2 \u2192 y_1 = y_2 from\n enum_unique A t\n done\n done\n \u00b7 -- Proof of fcnl_on (invRel (enum A)) A\n show fcnl_on (invRel (enum A)) A from inv_enum_fcnl A\n done\n done\n done\n\nlemma bdd_subset_nat {A : Set Nat} {m s : Nat}\n (h1 : \u2200 n \u2208 A, n < m) (h2 : num_elts_below A m s) :\n I s \u223c A := Exists.intro (enum A) (bdd_subset_nat_match h1 h2)\n\nlemma enum_fcnl_of_unbdd {A : Set Nat} (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n fcnl_on (enum A) (Univ Nat) := by\n define\n by_induc\n \u00b7 -- Base Case\n assume h2 : 0 \u2208 Univ Nat\n exists_unique\n \u00b7 -- Existence\n obtain (n : Nat) (h3 : n \u2208 A \u2227 n \u2265 0) from h1 0\n obtain (s : Nat) (h4 : num_elts_below A (n + 1) s) from neb_exists A (n + 1)\n have h5 : \u2200 (t : Nat), t < s \u2192 \u2203 (n : Nat), enum A t n := enum_not_skip h4\n rewrite [neb_step_elt h3.left] at h4\n show \u2203 (y : Nat), enum A 0 y from h5 0 h4.left\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A 0 y_1 \u2192 enum A 0 y_2 \u2192 y_1 = y_2 from enum_unique A 0\n done\n done\n \u00b7 -- Induction Step\n fix s : Nat\n assume ih : s \u2208 Univ Nat \u2192 \u2203! (y : Nat), enum A s y\n assume h2 : s + 1 \u2208 Univ Nat\n exists_unique\n \u00b7 -- Existence\n have h3 : s \u2208 Univ Nat := elt_Univ s\n obtain (m : Nat) (h4 : enum A s m)\n (h5 : \u2200 (y_1 y_2 : Nat), enum A s y_1 \u2192 enum A s y_2 \u2192 y_1 = y_2) from ih h3\n obtain (n : Nat) (h6 : n \u2208 A \u2227 n \u2265 m + 1) from h1 (m + 1)\n obtain (t : Nat) (h7 : num_elts_below A n t) from neb_exists A n\n have h8 : s < t := neb_increase h4 h6.right h7\n have h9 : s + 1 < t \u2228 s + 1 = t := Nat.lt_or_eq_of_le h8\n by_cases on h9\n \u00b7 -- Case 1. h9 : s + 1 < t\n show \u2203 (y : Nat), enum A (s + 1) y from enum_not_skip h7 (s + 1) h9\n done\n \u00b7 -- Case 2. h9 : s + 1 = t\n rewrite [h9]\n apply Exists.intro n\n define\n show n \u2208 A \u2227 num_elts_below A n t from And.intro h6.left h7\n done\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A (s + 1) y_1 \u2192 enum A (s + 1) y_2 \u2192 y_1 = y_2\n from enum_unique A (s + 1)\n done\n done\n done\n\nlemma unbdd_subset_nat_match {A : Set Nat}\n (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n matching (enum A) (Univ Nat) A := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix s : Nat; fix n : Nat\n assume h2 : enum A s n\n define at h2\n apply And.intro (elt_Univ s) h2.left\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on (enum A)\n show fcnl_on (enum A) (Univ Nat) from enum_fcnl_of_unbdd h1\n done\n \u00b7 -- Proof of fcnl_on (invRel (enum A))\n show fcnl_on (invRel (enum A)) A from inv_enum_fcnl A\n done\n done\n done\n\nlemma unbdd_subset_nat {A : Set Nat}\n (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n denum A := Exists.intro (enum A) (unbdd_subset_nat_match h1)\n\nlemma subset_nat_ctble (A : Set Nat) : ctble A := by\n define --Goal : finite A \u2228 denum A\n by_cases h1 : \u2203 (m : Nat), \u2200 n \u2208 A, n < m\n \u00b7 -- Case 1. h1 : \u2203 (m : Nat), \u2200 n \u2208 A, n < m\n apply Or.inl --Goal : finite A\n obtain (m : Nat) (h2 : \u2200 n \u2208 A, n < m) from h1\n obtain (s : Nat) (h3 : num_elts_below A m s) from neb_exists A m\n apply Exists.intro s\n show I s \u223c A from bdd_subset_nat h2 h3\n done\n \u00b7 -- Case 2. h1 : \u00ac\u2203 (m : Nat), \u2200 n \u2208 A, n < m\n apply Or.inr --Goal : denum A\n push_neg at h1\n --This tactic converts h1 to \u2200 (m : Nat), \u2203 n \u2208 A, m \u2264 n\n show denum A from unbdd_subset_nat h1\n done\n done\n\nlemma ctble_of_equinum_ctble {U V : Type} {A : Set U} {B : Set V}\n (h1 : A \u223c B) (h2 : ctble A) : ctble B := sorry\n\nlemma ctble_iff_equinum_set_nat {U : Type} (A : Set U) :\n ctble A \u2194 \u2203 (I : Set Nat), I \u223c A := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : ctble A\n define at h1 --h1 : finite A \u2228 denum A\n by_cases on h1\n \u00b7 -- Case 1. h1 : finite A\n define at h1 --h1 : \u2203 (n : Nat), I n \u223c A\n obtain (n : Nat) (h2 : I n \u223c A) from h1\n show \u2203 (I : Set Nat), I \u223c A from Exists.intro (I n) h2\n done\n \u00b7 -- Case 2. h1 : denum A\n rewrite [denum_def] at h1 --h1 : Univ Nat \u223c A\n show \u2203 (I : Set Nat), I \u223c A from Exists.intro (Univ Nat) h1\n done\n done\n \u00b7 -- (\u2190)\n assume h1 : \u2203 (I : Set Nat), I \u223c A\n obtain (I : Set Nat) (h2 : I \u223c A) from h1\n have h3 : ctble I := subset_nat_ctble I\n show ctble A from ctble_of_equinum_ctble h2 h3\n done\n done\n\ntheorem Theorem_8_1_5_1_to_2 {U : Type} {A : Set U} (h1 : ctble A) :\n \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A := by\n rewrite [ctble_iff_equinum_set_nat] at h1\n obtain (I : Set Nat) (h2 : I \u223c A) from h1\n obtain (R : Rel Nat U) (h3 : matching R I A) from h2\n define at h3\n --h3 : rel_within R I A \u2227 fcnl_on R I \u2227 fcnl_on (invRel R) A\n apply Exists.intro R\n define --Goal : unique_val_on_N R \u2227 nat_rel_onto R A\n apply And.intro\n \u00b7 -- Proof of unique_val_on_N R\n define\n fix n : Nat; fix x1 : U; fix x2 : U\n assume h4 : R n x1\n assume h5 : R n x2 --Goal : x1 = x2\n have h6 : n \u2208 I \u2227 x1 \u2208 A := h3.left h4\n show x1 = x2 from fcnl_unique h3.right.left h6.left h4 h5\n done\n \u00b7 -- Proof of nat_rel_onto R A\n define\n fix x : U\n assume h4 : x \u2208 A --Goal : \u2203 (n : Nat), R n x\n show \u2203 (n : Nat), R n x from fcnl_exists h3.right.right h4\n done\n done\n\nlemma exists_least_rel_to {U : Type} {S : Rel Nat U} {x : U}\n (h1 : \u2203 (n : Nat), S n x) : \u2203 (n : Nat), least_rel_to S x n := by\n set W : Set Nat := {n : Nat | S n x}\n have h2 : \u2203 (n : Nat), n \u2208 W := h1\n show \u2203 (n : Nat), least_rel_to S x n from well_ord_princ W h2\n done\n\n", "theoremStatement": "theorem Theorem_8_1_5_2_to_3 {U : Type} {A : Set U}\n (h1 : \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A) :\n \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A ", "theoremName": "HTPI.Theorem_8_1_5_2_to_3", "fileCreated": {"commit": "fe2721d65e0771872fe4cb8117517e643ac1dc1f", "date": 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"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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6", "HTPILib.Chap7"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain (S : Rel Nat U) (h2 : fcnl_onto_from_nat S A) from h1\n define at h2 --h2 : unique_val_on_N S \u2227 nat_rel_onto S A\n set R : Rel U Nat := least_rel_to S\n apply Exists.intro R\n define\n apply And.intro\n \u00b7 -- Proof of fcnl_on R A\n define\n fix x : U\n assume h4 : x \u2208 A --Goal : \u2203! (y : Nat), R x y\n exists_unique\n \u00b7 -- Existence\n have h5 : \u2203 (n : Nat), S n x := h2.right h4\n show \u2203 (n : Nat), R x n from exists_least_rel_to h5\n done\n \u00b7 -- Uniqueness\n fix n1 : Nat; fix n2 : Nat\n assume h5 : R x n1\n assume h6 : R x n2 --Goal : n1 = n2\n define at h5 --h5 : S n1 x \u2227 \u2200 (m : Nat), S m x \u2192 n1 \u2264 m\n define at h6 --h6 : S n2 x \u2227 \u2200 (m : Nat), S m x \u2192 n2 \u2264 m\n have h7 : n1 \u2264 n2 := h5.right n2 h6.left\n have h8 : n2 \u2264 n1 := h6.right n1 h5.left\n linarith\n done\n done\n \u00b7 -- Proof of one-to-one\n fix x1 : U; fix x2 : U; fix n : Nat\n assume h4 : x1 \u2208 A \u2227 R x1 n\n assume h5 : x2 \u2208 A \u2227 R x2 n\n have h6 : R x1 n := h4.right\n have h7 : R x2 n := h5.right\n define at h6 --h6 : S n x1 \u2227 \u2200 (m : Nat), S m x1 \u2192 n \u2264 m\n define at h7 --h7 : S n x2 \u2227 \u2200 (m : Nat), S m x2 \u2192 n \u2264 m\n show x1 = x2 from h2.left h6.left h7.left\n done\n done", "proofType": "tactic", "proofLengthLines": 37, "proofLengthTokens": 1248}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\n", "theoremStatement": "lemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p ", "theoremName": "HTPI.rel_prime_of_prime_not_dvd", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 568, "tokenPositionInFile": 17453, "theoremPositionInFile": 59}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 6, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 20, "importedModules": ["Init.Prelude", 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"Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done", "proofType": "tactic", "proofLengthLines": 11, "proofLengthTokens": 316}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap7\nnamespace HTPI\n\n/- Definitions -/\ndef fnz (n : Nat) : Int := if 2 \u2223 n then \u2191(n / 2) else -\u2191((n + 1) / 2)\n\ndef fzn (a : Int) : Nat := if a \u2265 0 then 2 * Int.toNat a else 2 * Int.toNat (-a) - 1\n\ndef tri (k : Nat) : Nat := k * (k + 1) / 2\n\ndef fnnn (p : Nat \u00d7 Nat) : Nat := tri (p.1 + p.2) + p.1\n\ndef num_elts_below (A : Set Nat) (m s : Nat) : Prop :=\n match m with\n | 0 => s = 0\n | n + 1 => (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s)\n\ndef enum (A : Set Nat) (s n : Nat) : Prop := n \u2208 A \u2227 num_elts_below A n s\n\ndef unique_val_on_N {U : Type} (R : Rel Nat U) : Prop :=\n \u2200 \u2983n : Nat\u2984 \u2983x1 x2 : U\u2984, R n x1 \u2192 R n x2 \u2192 x1 = x2\n\ndef nat_rel_onto {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203 (n : Nat), R n x\n\ndef fcnl_onto_from_nat {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n unique_val_on_N R \u2227 nat_rel_onto R A\n\ndef fcnl_one_one_to_nat {U : Type} (R : Rel U Nat) (A : Set U) : Prop :=\n fcnl_on R A \u2227 \u2200 \u2983x1 x2 : U\u2984 \u2983n : Nat\u2984,\n (x1 \u2208 A \u2227 R x1 n) \u2192 (x2 \u2208 A \u2227 R x2 n) \u2192 x1 = x2\n\ndef least_rel_to {U : Type} (S : Rel Nat U) (x : U) (n : Nat) : Prop :=\n S n x \u2227 \u2200 (m : Nat), S m x \u2192 n \u2264 m\n\ndef restrict_to {U V : Type} (S : Rel U V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 S x y\n\ndef fqn (q : Rat) : Nat := fnnn (fzn q.num, q.den)\n\ndef Set_rp_below (m : Nat) : Set Nat := {n : Nat | rel_prime m n \u2227 n < m}\n\ndef Set_prod {U V : Type} (A : Set U) (B : Set V) : Set (U \u00d7 V) :=\n {(a, b) : U \u00d7 V | a \u2208 A \u2227 b \u2208 B}\n\nnotation:75 A:75 \" \u00d7\u209b \" B:75 => Set_prod A B\n\ndef Rel_prod {U V W X : Type} (R : Rel U V) (S : Rel W X)\n (p : U \u00d7 W) (q : V \u00d7 X) : Prop := R p.1 q.1 \u2227 S p.2 q.2\n\nnotation:75 R:75 \" \u00d7\u1d63 \" S:75 => Rel_prod R S\n\ndef qr (n a : Nat) : Nat \u00d7 Nat := (a / n, a % n)\n\ndef mod_mod (m n a : Nat) : Nat \u00d7 Nat := (a % m, a % n)\n\ndef enum_union_fam {U : Type}\n (F : Set (Set U)) (f : Set U \u2192 Rel Nat U) (R : Rel Nat (Set U))\n (n : Nat) (a : U) : Prop := \u2203 (p : Nat \u00d7 Nat), fnnn p = n \u2227\n \u2203 A \u2208 F, R p.1 A \u2227 (f A) p.2 a\n\ndef seq {U : Type} (A : Set U) : Set (List U) :=\n {l : List U | \u2200 x \u2208 l, x \u2208 A}\n\ndef seq_by_length {U : Type} (A : Set U) (n : Nat) : Set (List U) :=\n {l : List U | l \u2208 seq A \u2227 l.length = n}\n\ndef seq_cons (U : Type) (p : U \u00d7 (List U)) : List U := p.1 :: p.2\n\ndef sbl_set {U : Type} (A : Set U) : Set (Set (List U)) :=\n {S : Set (List U) | \u2203 (n : Nat), seq_by_length A n = S}\n\ndef rep_common_image\n {U V : Type} (R S : Rel U V) (X0 : Set U) (n : Nat) : Set U :=\n match n with\n | 0 => X0\n | m + 1 => {a : U | \u2203 x \u2208 rep_common_image R S X0 m,\n \u2203 (y : V), R x y \u2227 S a y}\n\ndef cum_rep_image {U V : Type} (R S : Rel U V) (X0 : Set U) : Set U :=\n {a : U | \u2203 (n : Nat), a \u2208 rep_common_image R S X0 n}\n\ndef csb_match {U V : Type} (R S : Rel U V) (X0 : Set U)\n (x : U) (y : V) : Prop := x \u2208 cum_rep_image R S X0 \u2227 R x y \u2228\n x \u2209 cum_rep_image R S X0 \u2227 S x y\n\n/- Section 8.1 -/\n#eval [fnz 0, fnz 1, fnz 2, fnz 3, fnz 4, fnz 5, fnz 6]\n --Answer: [0, -1, 1, -2, 2, -3, 3]\n\n#eval [fzn 0, fzn (-1), fzn 1, fzn (-2), fzn 2, fzn (-3), fzn 3]\n --Answer: [0, 1, 2, 3, 4, 5, 6]\n\nlemma fnz_even (k : Nat) : fnz (2 * k) = \u2191k := by\n have h1 : 2 \u2223 2 * k := by\n apply Exists.intro k\n rfl\n done\n have h2 : fnz (2 * k) = if 2 \u2223 2 * k then \u2191(2 * k / 2)\n else -\u2191((2 * k + 1) / 2) := by rfl\n rewrite [if_pos h1] at h2 --h2 : fnz (2 * k) = \u2191(2 * k / 2)\n have h3 : 0 < 2 := by linarith\n rewrite [Nat.mul_div_cancel_left k h3] at h2\n show fnz (2 * k) = \u2191k from h2\n done\n\nlemma fnz_odd (k : Nat) : fnz (2 * k + 1) = -\u2191(k + 1) := sorry\n\nlemma fzn_nat (k : Nat) : fzn \u2191k = 2 * k := by rfl\n\nlemma fzn_neg_succ_nat (k : Nat) : fzn (-\u2191(k + 1)) = 2 * k + 1 := by rfl\n\n--From exercises of Section 6.1\ntheorem Exercise_6_1_16a1 : \u2200 (n : Nat), nat_even n \u2228 nat_odd n := sorry\n\nlemma fzn_fnz : fzn \u2218 fnz = id := by\n apply funext --Goal : \u2200 (x : Nat), (fzn \u2218 fnz) x = id x\n fix n : Nat\n rewrite [comp_def] --Goal : fzn (fnz n) = id n\n have h1 : nat_even n \u2228 nat_odd n := Exercise_6_1_16a1 n\n by_cases on h1\n \u00b7 -- Case 1. h1 : nat_even n\n obtain (k : Nat) (h2 : n = 2 * k) from h1\n rewrite [h2, fnz_even, fzn_nat]\n rfl\n done\n \u00b7 -- Case 2. h1 : nat_odd n\n obtain (k : Nat) (h2 : n = 2 * k + 1) from h1\n rewrite [h2, fnz_odd, fzn_neg_succ_nat]\n rfl\n done\n done\n\nlemma fnz_fzn : fnz \u2218 fzn = id := sorry\n\nlemma fzn_one_one : one_to_one fzn := Theorem_5_3_3_1 fzn fnz fnz_fzn\n\nlemma fzn_onto : onto fzn := Theorem_5_3_3_2 fzn fnz fzn_fnz\n\nlemma fnz_one_one : one_to_one fnz := Theorem_5_3_3_1 fnz fzn fzn_fnz\n\nlemma fnz_onto : onto fnz := Theorem_5_3_3_2 fnz fzn fnz_fzn\n\nlemma fnnn_def (a b : Nat) : fnnn (a, b) = tri (a + b) + a := by rfl\n\n#eval [fnnn (0, 0), fnnn (0, 1), fnnn (1, 0), fnnn (0, 2), fnnn (1, 1)]\n --Answer: [0, 1, 2, 3, 4]\n\nlemma tri_step (k : Nat) : tri (k + 1) = tri k + k + 1 := sorry\n\nlemma tri_incr {j k : Nat} (h1 : j \u2264 k) : tri j \u2264 tri k := sorry\n\nlemma le_of_fnnn_eq {a1 b1 a2 b2 : Nat}\n (h1 : fnnn (a1, b1) = fnnn (a2, b2)) : a1 + b1 \u2264 a2 + b2 := by\n by_contra h2\n have h3 : a2 + b2 + 1 \u2264 a1 + b1 := by linarith\n have h4 : fnnn (a2, b2) < fnnn (a1, b1) :=\n calc fnnn (a2, b2)\n _ = tri (a2 + b2) + a2 := by rfl\n _ < tri (a2 + b2) + (a2 + b2) + 1 := by linarith\n _ = tri (a2 + b2 + 1) := (tri_step _).symm\n _ \u2264 tri (a1 + b1) := tri_incr h3\n _ \u2264 tri (a1 + b1) + a1 := by linarith\n _ = fnnn (a1, b1) := by rfl\n linarith\n done\n\nlemma fnnn_one_one : one_to_one fnnn := by\n fix (a1, b1) : Nat \u00d7 Nat\n fix (a2, b2) : Nat \u00d7 Nat\n assume h1 : fnnn (a1, b1) = fnnn (a2, b2) --Goal : (a1, b1) = (a2, b2)\n have h2 : a1 + b1 \u2264 a2 + b2 := le_of_fnnn_eq h1\n have h3 : a2 + b2 \u2264 a1 + b1 := le_of_fnnn_eq h1.symm\n have h4 : a1 + b1 = a2 + b2 := by linarith\n rewrite [fnnn_def, fnnn_def, h4] at h1\n --h1 : tri (a2 + b2) + a1 = tri (a2 + b2) + a2\n have h6 : a1 = a2 := Nat.add_left_cancel h1\n rewrite [h6] at h4 --h4 : a2 + b1 = a2 + b2\n have h7 : b1 = b2 := Nat.add_left_cancel h4\n rewrite [h6, h7]\n rfl\n done\n\n", "theoremStatement": "lemma fnnn_onto : onto fnnn ", "theoremName": "HTPI.fnnn_onto", "fileCreated": {"commit": "fe2721d65e0771872fe4cb8117517e643ac1dc1f", "date": "2023-05-10"}, "theoremCreated": {"commit": "01538f73db4cc69cc47bd51e795bf25d251c4b6d", "date": "2023-05-31"}, "file": "htpi/HTPILib/Chap8Part2.lean", "module": "HTPILib.Chap8Part2", 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"Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6", "HTPILib.Chap7"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n define --Goal : \u2200 (y : Nat), \u2203 (x : Nat \u00d7 Nat), fnnn x = y\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro (0, 0)\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2203 (x : Nat \u00d7 Nat), fnnn x = n\n obtain ((a, b) : Nat \u00d7 Nat) (h1 : fnnn (a, b) = n) from ih\n by_cases h2 : b = 0\n \u00b7 -- Case 1. h2 : b = 0\n apply Exists.intro (0, a + 1)\n show fnnn (0, a + 1) = n + 1 from\n calc fnnn (0, a + 1)\n _ = tri (0 + (a + 1)) + 0 := by rfl\n _ = tri (a + 1) := by ring\n _ = tri a + a + 1 := tri_step a\n _ = tri (a + 0) + a + 1 := by ring\n _ = fnnn (a, b) + 1 := by rw [h2, fnnn_def]\n _ = n + 1 := by rw [h1]\n done\n \u00b7 -- Case 2. h2 : b \u2260 0\n obtain (k : Nat) (h3 : b = k + 1) from\n exists_eq_add_one_of_ne_zero h2\n apply Exists.intro (a + 1, k)\n show fnnn (a + 1, k) = n + 1 from\n calc fnnn (a + 1, k)\n _ = tri (a + 1 + k) + (a + 1) := by rfl\n _ = tri (a + (k + 1)) + a + 1 := by ring\n _ = tri (a + b) + a + 1 := by rw [h3]\n _ = fnnn (a, b) + 1 := by rfl\n _ = n + 1 := by rw [h1]\n done\n done\n done", "proofType": "tactic", "proofLengthLines": 36, "proofLengthTokens": 1183}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap5\nnamespace HTPI\n\n/- Definitions -/\ndef invRel {U V : Type} (R : Rel U V) : Rel V U :=\n RelFromExt (inv (extension R))\n\ndef rel_within {U V : Type} (R : Rel U V) (A : Set U) (B : Set V) : Prop :=\n \u2200 \u2983x : U\u2984 \u2983y : V\u2984, R x y \u2192 x \u2208 A \u2227 y \u2208 B\n\ndef fcnl_on {U V : Type} (R : Rel U V) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203! (y : V), R x y\n\ndef matching {U V : Type} (R : Rel U V) (A : Set U) (B : Set V) : Prop :=\n rel_within R A B \u2227 fcnl_on R A \u2227 fcnl_on (invRel R) B\n\ndef equinum {U V : Type} (A : Set U) (B : Set V) : Prop :=\n \u2203 (R : Rel U V), matching R A B\n\nnotation:50 A:50 \" \u223c \" B:50 => equinum A B\n\ndef RelWithinFromFunc {U V : Type} (f : U \u2192 V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 f x = y\n\ndef one_one_on {U V : Type} (f : U \u2192 V) (A : Set U) : Prop :=\n \u2200 \u2983x1 x2 : U\u2984, x1 \u2208 A \u2192 x2 \u2208 A \u2192 f x1 = f x2 \u2192 x1 = x2\n\ndef Univ (U : Type) : Set U := {x : U | True}\n\ndef compRel {U V W : Type} (S : Rel V W) (R : Rel U V) : Rel U W :=\n RelFromExt (comp (extension S) (extension R))\n\ndef I (n : Nat) : Set Nat := {k : Nat | k < n}\n\ndef finite {U : Type} (A : Set U) : Prop :=\n \u2203 (n : Nat), I n \u223c A\n\ndef denum {U : Type} (A : Set U) : Prop :=\n Univ Nat \u223c A\n\ndef ctble {U : Type} (A : Set U) : Prop :=\n finite A \u2228 denum A\n\ndef numElts {U : Type} (A : Set U) (n : Nat) : Prop := I n \u223c A\n\ndef emptyRel (U V : Type) (x : U) (y : V) : Prop := False\n\ndef remove_one {U V : Type} (R : Rel U V) (u : U) (v : V)\n (x : U) (y : V) : Prop := x \u2260 u \u2227 y \u2260 v \u2227 (R x y \u2228 (R x v \u2227 R u y))\n\ndef one_match {U V : Type} (a : U) (b : V)\n (x : U) (y : V) : Prop := x = a \u2227 y = b\n\n/- Section 8.1 -/\nlemma invRel_def {U V : Type} (R : Rel U V) (u : U) (v : V) :\n invRel R v u \u2194 R u v := by rfl\n\ntheorem equinum_image {U V : Type} {A : Set U} {B : Set V} {f : U \u2192 V}\n (h1 : one_one_on f A) (h2 : image f A = B) : A \u223c B := by\n rewrite [\u2190h2]\n define --Goal : \u2203 (R : Rel U V), matching R A (image f A)\n set R : Rel U V := RelWithinFromFunc f A\n apply Exists.intro R\n define --Goal : rel_within R A (image f A) \u2227\n --fcnl_on R A \u2227 fcnl_on (invRel R) (image f A)\n apply And.intro\n \u00b7 -- Proof of rel_within\n define --Goal : \u2200 \u2983x : U\u2984 \u2983y : V\u2984, R x y \u2192 x \u2208 A \u2227 y \u2208 image f A\n fix x : U; fix y : V\n assume h3 : R x y --Goal : x \u2208 A \u2227 y \u2208 image f A\n define at h3 --h3 : x \u2208 A \u2227 f x = y\n apply And.intro h3.left\n define\n show \u2203 x \u2208 A, f x = y from Exists.intro x h3\n done\n \u00b7 -- Proofs of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on R A\n define --Goal : \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203! (y : V), R x y\n fix x : U\n assume h3 : x \u2208 A\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (f x)\n define --Goal : x \u2208 A \u2227 f x = f x\n apply And.intro h3\n rfl\n done\n \u00b7 -- Uniqueness\n fix y1 : V; fix y2 : V\n assume h4 : R x y1\n assume h5 : R x y2 --Goal : y1 = y2\n define at h4; define at h5\n --h4 : x \u2208 A \u2227 f x = y1; h5 : x \u2208 A \u2227 f x = y2\n rewrite [h4.right] at h5\n show y1 = y2 from h5.right\n done\n done\n \u00b7 -- Proof of fcnl_on (invRel R) (image f A)\n define --Goal : \u2200 \u2983x : V\u2984, x \u2208 image f A \u2192 \u2203! (y : U), invRel R x y\n fix y : V\n assume h3 : y \u2208 image f A\n obtain (x : U) (h4 : x \u2208 A \u2227 f x = y) from h3\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro x\n define\n show x \u2208 A \u2227 f x = y from h4\n done\n \u00b7 -- Uniqueness\n fix x1 : U; fix x2 : U\n assume h5 : invRel R y x1\n assume h6 : invRel R y x2\n define at h5; define at h6\n --h5 : x1 \u2208 A \u2227 f x1 = y; h6 : x2 \u2208 A \u2227 f x2 = y\n rewrite [\u2190h6.right] at h5\n show x1 = x2 from h1 h5.left h6.left h5.right\n done\n done\n done\n done\n\nlemma id_one_one_on {U : Type} (A : Set U) : one_one_on id A := by\n define\n fix x1 : U; fix x2 : U\n assume h1 : x1 \u2208 A\n assume h2 : x2 \u2208 A\n assume h3 : id x1 = id x2\n show x1 = x2 from h3\n done\n\nlemma image_id {U : Type} (A : Set U) : image id A = A := by\n apply Set.ext\n fix x : U\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : x \u2208 image id A\n obtain (y : U) (h2 : y \u2208 A \u2227 id y = x) from h1\n rewrite [\u2190h2.right]\n show id y \u2208 A from h2.left\n done\n \u00b7 -- (\u2190)\n assume h1 : x \u2208 A\n apply Exists.intro x --Goal : x \u2208 A \u2227 id x = x\n apply And.intro h1\n rfl\n done\n done\n\ntheorem Theorem_8_1_3_1 {U : Type} (A : Set U) : A \u223c A :=\n equinum_image (id_one_one_on A) (image_id A)\n\nlemma inv_inv {U V : Type} (R : Rel U V) : invRel (invRel R) = R := by rfl\n\nlemma inv_match {U V : Type} {R : Rel U V} {A : Set U} {B : Set V}\n (h : matching R A B) : matching (invRel R) B A := by\n define --Goal : rel_within (invRel R) B A \u2227\n --fcnl_on (invRel R) B \u2227 fcnl_on (invRel (invRel R)) A\n define at h --h : rel_within R A B \u2227 fcnl_on R A \u2227 fcnl_on (invRel R) B\n apply And.intro\n \u00b7 -- Proof that rel_within (invRel R) B A\n define --Goal : \u2200 \u2983x : V\u2984 \u2983y : U\u2984, invRel R x y \u2192 x \u2208 B \u2227 y \u2208 A\n fix y : V; fix x : U\n assume h1 : invRel R y x\n define at h1 --h1 : R x y\n have h2 : x \u2208 A \u2227 y \u2208 B := h.left h1\n show y \u2208 B \u2227 x \u2208 A from And.intro h2.right h2.left\n done\n \u00b7 -- proof that fcnl_on (inv R) B \u2227 fcnl_on (inv (inv R)) A\n rewrite [inv_inv]\n show fcnl_on (invRel R) B \u2227 fcnl_on R A from\n And.intro h.right.right h.right.left\n done\n done\n\ntheorem Theorem_8_1_3_2 {U V : Type} {A : Set U} {B : Set V}\n (h : A \u223c B) : B \u223c A := by\n obtain (R : Rel U V) (h1 : matching R A B) from h\n apply Exists.intro (invRel R)\n show matching (invRel R) B A from inv_match h1\n done\n\nlemma fcnl_exists {U V : Type} {R : Rel U V} {A : Set U} {x : U}\n (h1 : fcnl_on R A) (h2 : x \u2208 A) : \u2203 (y : V), R x y := by\n define at h1\n obtain (y : V) (h3 : R x y)\n (h4 : \u2200 (y_1 y_2 : V), R x y_1 \u2192 R x y_2 \u2192 y_1 = y_2) from h1 h2\n show \u2203 (y : V), R x y from Exists.intro y h3\n done\n\nlemma fcnl_unique {U V : Type}\n {R : Rel U V} {A : Set U} {x : U} {y1 y2 : V} (h1 : fcnl_on R A)\n (h2 : x \u2208 A) (h3 : R x y1) (h4 : R x y2) : y1 = y2 := by\n define at h1\n obtain (z : V) (h5 : R x z)\n (h6 : \u2200 (y_1 y_2 : V), R x y_1 \u2192 R x y_2 \u2192 y_1 = y_2) from h1 h2\n show y1 = y2 from h6 y1 y2 h3 h4\n done\n\nlemma compRel_def {U V W : Type}\n (S : Rel V W) (R : Rel U V) (u : U) (w : W) :\n compRel S R u w \u2194 \u2203 (x : V), R u x \u2227 S x w := by rfl\n\nlemma inv_comp {U V W : Type} (R : Rel U V) (S : Rel V W) :\n invRel (compRel S R) = compRel (invRel R) (invRel S) :=\n calc invRel (compRel S R)\n _ = RelFromExt (inv (comp (extension S) (extension R))) := by rfl\n _ = RelFromExt (comp (inv (extension R)) (inv (extension S))) := by\n rw [Theorem_4_2_5_5]\n _ = compRel (invRel R) (invRel S) := by rfl\n\nlemma comp_fcnl {U V W : Type} {R : Rel U V} {S : Rel V W}\n {A : Set U} {B : Set V} {C : Set W} (h1 : matching R A B)\n (h2 : matching S B C) : fcnl_on (compRel S R) A := by\n define; define at h1; define at h2\n fix a : U\n assume h3 : a \u2208 A\n obtain (b : V) (h4 : R a b) from fcnl_exists h1.right.left h3\n have h5 : a \u2208 A \u2227 b \u2208 B := h1.left h4\n obtain (c : W) (h6 : S b c) from fcnl_exists h2.right.left h5.right\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro c\n rewrite [compRel_def]\n show \u2203 (x : V), R a x \u2227 S x c from Exists.intro b (And.intro h4 h6)\n done\n \u00b7 -- Uniqueness\n fix c1 : W; fix c2 : W\n assume h7 : compRel S R a c1\n assume h8 : compRel S R a c2 --Goal : c1 = c2\n rewrite [compRel_def] at h7\n rewrite [compRel_def] at h8\n obtain (b1 : V) (h9 : R a b1 \u2227 S b1 c1) from h7\n obtain (b2 : V) (h10 : R a b2 \u2227 S b2 c2) from h8\n have h11 : b1 = b := fcnl_unique h1.right.left h3 h9.left h4\n have h12 : b2 = b := fcnl_unique h1.right.left h3 h10.left h4\n rewrite [h11] at h9\n rewrite [h12] at h10\n show c1 = c2 from\n fcnl_unique h2.right.left h5.right h9.right h10.right\n done\n done\n\nlemma comp_match {U V W : Type} {R : Rel U V} {S : Rel V W}\n {A : Set U} {B : Set V} {C : Set W} (h1 : matching R A B)\n (h2 : matching S B C) : matching (compRel S R) A C := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within (compRel S R) A C\n define\n fix a : U; fix c : W\n assume h3 : compRel S R a c\n rewrite [compRel_def] at h3\n obtain (b : V) (h4 : R a b \u2227 S b c) from h3\n have h5 : a \u2208 A \u2227 b \u2208 B := h1.left h4.left\n have h6 : b \u2208 B \u2227 c \u2208 C := h2.left h4.right\n show a \u2208 A \u2227 c \u2208 C from And.intro h5.left h6.right\n done\n \u00b7 -- Proof of fcnl_on statements\n apply And.intro\n \u00b7 -- Proof of fcnl_on (compRel S R) A\n show fcnl_on (compRel S R) A from comp_fcnl h1 h2\n done\n \u00b7 -- Proof of fcnl_on (invRel (compRel S R)) Z\n rewrite [inv_comp]\n have h3 : matching (invRel R) B A := inv_match h1\n have h4 : matching (invRel S) C B := inv_match h2\n show fcnl_on (compRel (invRel R) (invRel S)) C from comp_fcnl h4 h3\n done\n done\n done\n\ntheorem Theorem_8_1_3_3 {U V W : Type} {A : Set U} {B : Set V} {C : Set W}\n (h1 : A \u223c B) (h2 : B \u223c C) : A \u223c C := by\n obtain (R : Rel U V) (h3 : matching R A B) from h1\n obtain (S : Rel V W) (h4 : matching S B C) from h2\n apply Exists.intro (compRel S R)\n show matching (compRel S R) A C from comp_match h3 h4\n done\n\nlemma I_def (k n : Nat) : k \u2208 I n \u2194 k < n := by rfl\n\nlemma denum_def {U : Type} (A : Set U) : denum A \u2194 Univ Nat \u223c A := by rfl\n\n/- Section 8.1\u00bd -/\nlemma numElts_def {U : Type} (A : Set U) (n : Nat) :\n numElts A n \u2194 I n \u223c A := by rfl\n\nlemma finite_def {U : Type} (A : Set U) :\n finite A \u2194 \u2203 (n : Nat), numElts A n := by rfl\n\nlemma fcnl_on_empty {U V : Type}\n (R : Rel U V) {A : Set U} (h1 : empty A) : fcnl_on R A := by\n define\n fix a : U\n assume h2 : a \u2208 A --Goal : \u2203! (y : V), R a y\n contradict h1 with h3 --Goal : \u2203 (x : U), x \u2208 A\n show \u2203 (x : U), x \u2208 A from Exists.intro a h2\n done\n\nlemma empty_match {U V : Type} {A : Set U} {B : Set V}\n (h1 : empty A) (h2 : empty B) : matching (emptyRel U V) A B := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix a : U; fix b : V\n assume h3 : emptyRel U V a b --Goal : a \u2208 A \u2227 b \u2208 B\n by_contra h4 --Goal : False\n define at h3\n show False from h3\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on emptyRel\n show fcnl_on (emptyRel U V) A from fcnl_on_empty (emptyRel U V) h1\n done\n \u00b7 -- Proof of fcnl_on (invRel emptyRel)\n show fcnl_on (invRel (emptyRel U V)) B from\n fcnl_on_empty (invRel (emptyRel U V)) h2\n done\n done\n\nlemma I_0_empty : empty (I 0) := by\n define\n by_contra h1 --h1 : \u2203 (x : Nat), x \u2208 I 0\n obtain (x : Nat) (h2 : x \u2208 I 0) from h1\n define at h2 --h2 : x < 0\n linarith\n done\n\ntheorem zero_elts_iff_empty {U : Type} (A : Set U) :\n numElts A 0 \u2194 empty A := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : numElts A 0\n define\n by_contra h2 --h2 : \u2203 (x : U), x \u2208 A\n obtain (x : U) (h3 : x \u2208 A) from h2\n define at h1\n obtain (R : Rel Nat U) (h4 : matching R (I 0) A) from h1\n define at h4\n --h4 : rel_within R (I 0) A \u2227 fcnl_on R (I 0) \u2227 fcnl_on (invRel R) A\n obtain (j : Nat) (h5 : invRel R x j) from\n fcnl_exists h4.right.right h3\n define at h5 --h5 : R j x\n have h6 : j \u2208 I 0 \u2227 x \u2208 A := h4.left h5\n contradict I_0_empty --Goal : \u2203 (x : Nat), x \u2208 I 0\n show \u2203 (x : Nat), x \u2208 I 0 from Exists.intro j h6.left\n done\n \u00b7 -- (\u2190)\n assume h1 : empty A\n show \u2203 (R : Rel Nat U), matching R (I 0) A from\n Exists.intro (emptyRel Nat U) (empty_match I_0_empty h1)\n done\n done\n\ntheorem nonempty_of_pos_numElts {U : Type} {A : Set U} {n : Nat}\n (h1 : numElts A n) (h2 : n > 0) : \u2203 (x : U), x \u2208 A := by\n define at h1\n obtain (R : Rel Nat U) (h3 : matching R (I n) A) from h1\n define at h3\n have h4 : 0 \u2208 I n := h2\n obtain (x : U) (h5 : R 0 x) from fcnl_exists h3.right.left h4\n have h6 : 0 \u2208 I n \u2227 x \u2208 A := h3.left h5\n show \u2203 (x : U), x \u2208 A from Exists.intro x h6.right\n done\n\ntheorem relext {U V : Type} {R S : Rel U V}\n (h : \u2200 (u : U) (v : V), R u v \u2194 S u v) : R = S := by\n have h2 : extension R = extension S := by\n apply Set.ext\n fix (u, v) : U \u00d7 V --Goal : (u, v) \u2208 extension R \u2194 (u, v) \u2208 extension S\n rewrite [ext_def, ext_def] --Goal : R u v \u2194 S u v\n show R u v \u2194 S u v from h u v\n done\n show R = S from\n calc R\n _ = RelFromExt (extension R) := by rfl\n _ = RelFromExt (extension S) := by rw [h2]\n _ = S := by rfl\n done\n\nlemma remove_one_def {U V : Type} (R : Rel U V) (u x : U) (v y : V) :\n remove_one R u v x y \u2194\n x \u2260 u \u2227 y \u2260 v \u2227 (R x y \u2228 (R x v \u2227 R u y)) := by rfl\n\nlemma remove_one_rel_within {U V : Type}\n {R : Rel U V} {A : Set U} {B : Set V} {x u : U} {y v : V}\n (h1 : matching R A B) (h2 : remove_one R u v x y) :\n x \u2208 A \\ {u} \u2227 y \u2208 B \\ {v} := by\n define at h1 --h1 : rel_within R A B \u2227 fcnl_on R A \u2227 fcnl_on (invRel R) B\n define at h2 --h2 : x \u2260 u \u2227 y \u2260 v \u2227 (R x y \u2228 R x v \u2227 R u y)\n have h3 : x \u2208 A \u2227 y \u2208 B := by\n by_cases on h2.right.right with h3\n \u00b7 -- Case 1. h3 : R x y\n show x \u2208 A \u2227 y \u2208 B from h1.left h3\n done\n \u00b7 -- Case 2. h3 : R x v \u2227 R u y\n have h4 : x \u2208 A \u2227 v \u2208 B := h1.left h3.left\n have h5 : u \u2208 A \u2227 y \u2208 B := h1.left h3.right\n show x \u2208 A \u2227 y \u2208 B from And.intro h4.left h5.right\n done\n done\n show x \u2208 A \\ {u} \u2227 y \u2208 B \\ {v} from\n And.intro (And.intro h3.left h2.left) (And.intro h3.right h2.right.left)\n done\n\nlemma remove_one_inv {U V : Type} (R : Rel U V) (u : U) (v : V) :\n invRel (remove_one R u v) = remove_one (invRel R) v u := by\n apply relext\n fix y : V; fix x : U\n --Goal : invRel (remove_one R u v) y x \u2194 remove_one (invRel R) v u y x\n rewrite [invRel_def, remove_one_def, remove_one_def]\n rewrite [invRel_def, invRel_def, invRel_def]\n rewrite [\u2190and_assoc, \u2190and_assoc]\n --Goal : (x \u2260 u \u2227 y \u2260 v) \u2227 (R x y \u2228 R x v \u2227 R u y) \u2194\n -- (y \u2260 v \u2227 x \u2260 u) \u2227 (R x y \u2228 R u y \u2227 R x v)\n have h1 : x \u2260 u \u2227 y \u2260 v \u2194 y \u2260 v \u2227 x \u2260 u := and_comm\n have h2 : R x v \u2227 R u y \u2194 R u y \u2227 R x v := and_comm\n rewrite [h1, h2]\n rfl\n done\n\nlemma remove_one_iff {U V : Type}\n {A : Set U} {B : Set V} {R : Rel U V} (h1 : matching R A B)\n {u : U} (h2 : u \u2208 A) (v : V) {x : U} (h3 : x \u2208 A \\ {u}) :\n \u2203 w \u2208 A, \u2200 (y : V), remove_one R u v x y \u2194 R w y := sorry\n\ntheorem remove_one_fcnl {U V : Type}\n {R : Rel U V} {A : Set U} {B : Set V} {u : U}\n (h1 : matching R A B) (h2 : u \u2208 A) (v : V) :\n fcnl_on (remove_one R u v) (A \\ {u}) := by\n define\n fix x : U\n assume h3 : x \u2208 A \\ {u} --Goal : \u2203! (y : V), remove_one R u v x y\n obtain (w : U) (h4 : w \u2208 A \u2227 \u2200 (y : V),\n remove_one R u v x y \u2194 R w y) from remove_one_iff h1 h2 v h3\n define at h1\n exists_unique\n \u00b7 -- Existence\n obtain (y : V) (h5 : R w y) from fcnl_exists h1.right.left h4.left\n apply Exists.intro y\n rewrite [h4.right]\n show R w y from h5\n done\n \u00b7 -- Uniqueness\n fix y1 : V; fix y2 : V\n rewrite [h4.right, h4.right]\n assume h5 : R w y1\n assume h6 : R w y2\n show y1 = y2 from fcnl_unique h1.right.left h4.left h5 h6\n done\n done\n\ntheorem remove_one_match {U V : Type}\n {R : Rel U V} {A : Set U} {B : Set V} {u : U} {v : V}\n (h1 : matching R A B) (h2 : u \u2208 A) (h3 : v \u2208 B) :\n matching (remove_one R u v) (A \\ {u}) (B \\ {v}) := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix x : U; fix y : V\n assume h4 : remove_one R u v x y\n show x \u2208 A \\ {u} \u2227 y \u2208 B \\ {v} from remove_one_rel_within h1 h4\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro (remove_one_fcnl h1 h2 v)\n rewrite [remove_one_inv]\n show fcnl_on (remove_one (invRel R) v u) (B \\ {v}) from\n remove_one_fcnl (inv_match h1) h3 u\n done\n\ntheorem remove_one_equinum {U V : Type}\n {A : Set U} {B : Set V} {u : U} {v : V}\n (h1 : A \u223c B) (h2 : u \u2208 A) (h3 : v \u2208 B) : A \\ {u} \u223c B \\ {v} := by\n define\n obtain (R : Rel U V) (h4 : matching R A B) from h1\n apply Exists.intro (remove_one R u v)\n show matching (remove_one R u v) (A \\ {u}) (B \\ {v}) from\n remove_one_match h4 h2 h3\n done\n\nlemma I_max (n : Nat) : n \u2208 I (n + 1) := by\n define\n linarith\n done\n\nlemma I_diff (n : Nat) : I (n + 1) \\ {n} = I n := by\n apply Set.ext\n fix x : Nat\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : x \u2208 I (n + 1) \\ {n}\n define --Goal : x < n\n define at h1\n have h2 : x \u2208 I (n + 1) := h1.left\n have h3 : \u00acx \u2208 {n} := h1.right\n define at h2 --h2 : x < n + 1\n define at h3 --h3 : \u00acx = n\n have h4 : x \u2264 n := Nat.le_of_lt_succ h2\n show x < n from Nat.lt_of_le_of_ne h4 h3\n done\n \u00b7 -- (\u2190)\n assume h1 : x \u2208 I n\n define at h1 --h1 : x < n\n define --Goal : x \u2208 I (n + 1) \u2227 \u00acx \u2208 {n}\n apply And.intro\n \u00b7 -- Proof that x \u2208 I (n + 1)\n define --Goal : x < n + 1\n linarith\n done\n \u00b7 -- Proof that x \u2209 {n}\n by_contra h2 --h2 : x \u2208 {n}\n define at h2 --h2 : x = n\n linarith\n done\n done\n done\n\ntheorem remove_one_numElts {U : Type} {A : Set U} {n : Nat} {a : U}\n (h1 : numElts A (n + 1)) (h2 : a \u2208 A) : numElts (A \\ {a}) n := by\n have h3 : n \u2208 I (n + 1) := I_max n\n rewrite [numElts_def] at h1 --h1 : I (n + 1) \u223c X\n have h4 : I (n + 1) \\ {n} \u223c A \\ {a} := remove_one_equinum h1 h3 h2\n rewrite [I_diff] at h4 --h4 : I n \u223c A \\ {a}\n show numElts (A \\ {a}) n from h4\n done\n\nlemma one_match_def {U V : Type} (a x : U) (b y : V) :\n one_match a b x y \u2194 x = a \u2227 y = b := by rfl\n\nlemma one_match_match {U V : Type} (a : U) (b : V) :\n matching (one_match a b) {a} {b} := sorry\n\nlemma I_1_singleton : I 1 = {0} := by\n apply Set.ext\n fix x : Nat\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : x \u2208 I 1\n rewrite [I_def] at h1 --h1 : x < 1\n define --Goal : x = 0\n linarith\n done\n \u00b7 -- (\u2190)\n assume h1 : x \u2208 {0}\n define at h1 --h1 : x = 0\n rewrite [h1, I_def] --Goal : 0 < 1\n linarith\n done\n done\n\nlemma singleton_of_diff_empty {U : Type} {A : Set U} {a : U}\n (h1 : a \u2208 A) (h2 : empty (A \\ {a})) : A = {a} := by\n apply Set.ext\n fix x : U\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h3 : x \u2208 A\n contradict h2 with h4\n apply Exists.intro x\n define\n show x \u2208 A \u2227 \u00acx \u2208 {a} from And.intro h3 h4\n done\n \u00b7 -- (\u2190)\n assume h3 : x \u2208 {a}\n define at h3\n rewrite [h3]\n show a \u2208 A from h1\n done\n done\n\n", "theoremStatement": "lemma singleton_one_elt {U : Type} (u : U) : numElts 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"Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n define\n rewrite [I_1_singleton]\n show \u2203 (R : Rel Nat U), matching R {0} {u} from\n Exists.intro (one_match 0 u) (one_match_match 0 u)\n done", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 152}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport Lean.Elab.Tactic\nimport Mathlib.Tactic\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Data.Set.Lattice\nimport Mathlib.Data.Rel\nimport Mathlib.Data.Nat.Defs\nimport Mathlib.Data.Int.Defs\nimport Mathlib.Data.Rat.Defs\nimport Mathlib.Data.Real.Basic\nimport Mathlib.Data.ZMod.Defs\n\ndef Iff.ltr {p q : Prop} (h : p \u2194 q) := h.mp\ndef Iff.rtl {p q : Prop} (h : p \u2194 q) := h.mpr\n\n--Make sure Lean understands {x} and \u2205 as Sets, not Finsets\nattribute [default_instance] Set.instSingletonSet\nattribute [default_instance] Set.instEmptyCollectionSet\n\n-- Used in one exercise in Chapter 3.\nnotation:50 a:50 \" \u2288 \" b:50 => \u00ac (a \u2286 b)\n\n--Note: Mathlib.Order.SymmDiff.lean defines this, but it is scoped[symmDiff] there.\n-- Use in one exercise in Chapter 3.\ninfixl:100 \" \u2206 \" => symmDiff\n\nnamespace HTPI\n--Some theorems not in library\ntheorem not_not_and_distrib {p q : Prop} : \u00ac(\u00ac p \u2227 q) \u2194 (p \u2228 \u00ac q) := by\n rw [not_and_or, Classical.not_not]\n\ntheorem not_and_not_distrib {p q : Prop} : \u00ac(p \u2227 \u00ac q) \u2194 (\u00ac p \u2228 q) := by\n rw [not_and_or, Classical.not_not]\n\ntheorem not_not_or_distrib {p q : Prop} : \u00ac(\u00ac p \u2228 q) \u2194 (p \u2227 \u00ac q) := by\n rw [not_or, Classical.not_not]\n\ntheorem not_or_not_distrib {p q : Prop} : \u00ac(p \u2228 \u00ac q) \u2194 (\u00ac p \u2227 q) := by\n rw [not_or, Classical.not_not]\n\ntheorem not_imp_not_iff_and {p q : Prop} : \u00ac (p \u2192 \u00ac q) \u2194 p \u2227 q := by\n rw [not_imp, Classical.not_not]\n\ntheorem not_imp_iff_not_and {p q : Prop} : \u00ac (q \u2192 p) \u2194 \u00ac p \u2227 q := by\n rw [not_imp]\n exact And.comm\n\ntheorem not_not_iff {p q : Prop} : \u00ac(\u00acp \u2194 q) \u2194 (p \u2194 q) := by\n rw [not_iff, Classical.not_not]\n\ndef Pred (t : Type u) : Type u := t \u2192 Prop\ndef BinRel (t : Type u) : Type u := Rel t t\n\n--Definitions of tactics\nsection tactic_defs\nopen Lean Elab Tactic Expr MVarId\n\n--Syntax for arguments to tactics\nsyntax oneLoc := \" at \" ident\nsyntax colonTerm := \" : \" term\nsyntax withId := \" with \" ident\nsyntax with2Ids := \" with \" ident (\", \" ident)?\nsyntax idOrTerm := ident <|> (\"(\" term \")\")\nsyntax idOrTerm?Type := ident <|> (\"(\" term (\" : \" term)? \")\")\n\nabbrev OneLoc := TSyntax ``oneLoc\nabbrev ColonTerm := TSyntax ``colonTerm\nabbrev WithId := TSyntax ``withId\nabbrev With2Ids := TSyntax ``with2Ids\nabbrev IdOrTerm := TSyntax ``idOrTerm\nabbrev IdOrTerm?Type := TSyntax ``idOrTerm?Type\n\n--Get formula from identifier\ndef formFromIdent (h : Syntax) : TacticM Expr := do\n instantiateMVars (\u2190 Meta.getLocalDeclFromUserName h.getId).type\n\n--Get formula from optional location. Note both formFromIdent and getMainTarget call instantiateMVars\ndef formFromLoc (l : Option OneLoc) : TacticM Expr := do\n match l with\n | some h => formFromIdent h.raw[1]\n | none => getMainTarget\n\n--For debugging:\ndef myTrace (msg : String) : TacticM Unit := do\n let m := Syntax.mkStrLit msg\n evalTactic (\u2190 `(tactic| trace $m))\n\npartial def SyntaxToString (s : Syntax) : String :=\nmatch s with\n | .missing => \"(missing)\"\n | .node _ k as => \"(node \" ++ toString k ++ (SyntaxListToString as.data) ++ \")\"\n | .atom _ v => \"(atom \" ++ toString v ++ \")\"\n | .ident _ rv v _ => \"(ident \" ++ (toString rv) ++ \" \" ++ (toString v) ++ \")\"\nwhere SyntaxListToString (ss : List Syntax) : String :=\n match ss with\n | (s :: rest) => \" \" ++ (SyntaxToString s) ++ (SyntaxListToString rest)\n | [] => \"\"\n\ndef traceThisSyntax (s : Syntax) : TacticM Unit := myTrace (SyntaxToString s)\n\ndef binderString (bi : BinderInfo) : String :=\n match bi with\n | .default => \"default\"\n | _ => \"not default\"\n\ndef ExprToString (e : Expr) : String :=\nmatch e with\n | .bvar n => \"(bvar \" ++ (toString n) ++ \")\" -- bound variables\n | .fvar f => \"(fvar \" ++ (toString f.name) ++ \")\" -- free variables\n | .mvar m => \"(mvar \" ++ (toString m.name) ++ \")\" -- meta variables\n | .sort l => \"(sort \" ++ (toString l) ++ \")\" -- Sort\n | .const n ls => \"(const \" ++ (toString n) ++ \" \" ++ (toString ls) ++ \")\" -- constants\n | .app a b => \"(app \" ++ (ExprToString a) ++ \" \" ++ (ExprToString b) ++ \")\" -- application\n | .lam n t b bi => \"(lam \" ++ (toString n) ++ \" \" ++ (ExprToString t) ++ \" \" ++ (ExprToString b) ++ \" \" ++ (binderString bi) ++ \")\" -- lambda abstraction\n | .forallE n t b bi => \"(forallE \" ++ (toString n) ++ \" \" ++ (ExprToString t) ++ \" \" ++ (ExprToString b) ++ \" \" ++ (binderString bi) ++ \")\" -- (dependent) arrow\n | .letE n t v b _ => \"(let \" ++ (toString n) ++ \" \" ++ (ExprToString t) ++ \" \"\n ++ (ExprToString v) ++ \" \" ++ (ExprToString b) ++ \")\" -- let expressions\n | .lit _ => \"(lit)\" -- literals\n | .mdata m e => \"(mdata \" ++ (toString m) ++ \" \" ++ (ExprToString e) ++ \")\" -- metadata\n | .proj t i c => \"(proj \" ++ (toString t) ++ \" \" ++ (toString i) ++ \" \" ++ (ExprToString c) ++ \")\" -- projection\n\ndef traceThisExpr (e : Expr) : TacticM Unit := myTrace (ExprToString e)\n\nelab \"traceExpr\" t:(colonTerm)? l:(oneLoc)? : tactic =>\n withMainContext do\n match t with\n | some tstx => do\n traceThisSyntax tstx.raw[1]\n let e \u2190 elabTerm tstx.raw[1] none\n traceThisExpr e\n | none =>\n let e \u2190 formFromLoc l\n traceThisExpr e\n\n-- Get head and arg list\ndef getHeadData (e : Expr) : Expr \u00d7 List Expr :=\n match e with\n | app f a =>\n let (h, as) := getHeadData f\n (h, a :: as)\n | mdata _ e' => getHeadData e'\n | _ => (e, [])\n\n-- Recover expression from head and arg list\ndef mkAppList (h : Expr) (args : List Expr) : Expr :=\n match args with\n | a :: rest => mkApp (mkAppList h rest) a\n | [] => h\n\n--Determine if e is a proposition, in current local context\ndef exprIsProp (e : Expr) : TacticM Bool :=\n return (\u2190 Meta.inferType e).isProp\n\n--Logical form of a proposition.\ninductive PropForm where\n | not : Expr \u2192 PropForm\n | and : Expr \u2192 Expr \u2192 PropForm\n | or : Expr \u2192 Expr \u2192 PropForm\n | implies : Expr \u2192 Expr \u2192 PropForm\n | iff : Expr \u2192 Expr \u2192 PropForm\n | all : Name \u2192 Expr \u2192 Expr \u2192 BinderInfo \u2192 PropForm\n | ex : Level \u2192 Name \u2192 Expr \u2192 Expr \u2192 BinderInfo \u2192 PropForm\n | exun : Level \u2192 Name \u2192 Expr \u2192 Expr \u2192 BinderInfo \u2192 PropForm\n | f : PropForm\n | t : PropForm\n | none : PropForm\n\n/- Try to unfold definition, and if result is negative, return PropForm.not\nNote: Uses constants but not fvars with let declarations. Also, only unfolds once.\nThis might be best--only detect expressions immediately recognized as negative by def.\n-/\ndef findNegPropAll (e : Expr) : TacticM PropForm := do\n match (\u2190 Meta.unfoldDefinition? (consumeMData e)) with\n | some e' =>\n match getHeadData e' with\n | (const ``Not _, [l]) => return PropForm.not l\n | _ => return PropForm.none\n | none => return PropForm.none\n\n--Apply a function to data for an existential. Existentials usually apply to a\n--lambda expression, but allow for others\ndef applyToExData {\u03b1 : Type} (f : Level \u2192 Name \u2192 Expr \u2192 Expr \u2192 BinderInfo \u2192 \u03b1)\n (lev : Level) (l r : Expr) : \u03b1 :=\n let r' := consumeMData r\n match r' with\n | lam v t b bi => f lev v t b bi\n | _ => f lev `x l (mkApp r' (bvar 0)) BinderInfo.default\n\n-- Get logical form of a proposition.\n-- Recognizes negative predicates by findNegPropAll.\ndef getPropForm (e : Expr) : TacticM PropForm := do\n if !(\u2190 exprIsProp e) then return PropForm.none\n let (h, args) := getHeadData e\n match h with\n | const c levs =>\n match (c, levs, args) with\n | (``False, _, _) => return PropForm.f\n | (``True, _, _) => return PropForm.t\n | (``Not, _, [l]) => return PropForm.not l\n | (``And, _, [r, l]) => return PropForm.and l r\n | (``Or, _, [r, l]) => return PropForm.or l r\n | (``Iff, _, [r, l]) => return PropForm.iff l r\n | (``Exists, [lev], [r, l]) => return applyToExData PropForm.ex lev l r\n | (``ExistsUnique, [lev], [r, l]) => return applyToExData PropForm.exun lev l r\n | _ => findNegPropAll e\n | forallE v t b bi =>\n if (b.hasLooseBVars || !(\u2190 exprIsProp t)) then\n return PropForm.all v t b bi\n else\n return PropForm.implies t b\n | _ => return PropForm.none\n\n--mkNot, mkAnd, mkOr, and mkForall are already defined. Also mkArrow and Meta.mkEq\ndef mkIff (l r : Expr) : Expr :=\n mkApp2 (mkConst ``Iff) l r\n\n--Need to supply level--I always have it, so easiest to use it.\ndef mkExists (l : Level) (x : Name) (bi : BinderInfo) (t b : Expr) : Expr :=\n mkApp2 (mkConst ``Exists [l]) t (mkLambda x bi t b)\n\ndef myFail {\u03b1} (tac : Name) (msg : String) : TacticM \u03b1 := do\n Meta.throwTacticEx tac (\u2190 getMainGoal) msg\n\n/- Functions for unfolding head -/\n\n--Unfold ExistsUnique; default version doesn't do a good job of naming variables\ndef unfoldExUn (lev : Level) (v : Name) (t b : Expr) (_ : BinderInfo) : Expr :=\n let v1 := Name.appendIndexAfter v 1\n let eqn := mkApp3 (mkConst ``Eq [lev]) t (bvar 1) (bvar 2)\n let body := mkAnd b (mkForall v1 BinderInfo.default t (mkForall `x BinderInfo.default b eqn))\n mkExists lev v BinderInfo.default t body\n\n-- Constants not to unfold except if explicitly asked to\ndef dontUnfold : List Name := [``ite, ``dite]\ndef dontUnfoldNum : List Name := [``LT.lt, ``LE.le, ``GT.gt, ``GE.ge]\ndef numNames : List Name := [``Nat, ``Int, ``_root_.Rat, ``Real]\n\n/- Unfold head in current context--must set local context before call.\nIf first = true, then unfold ExistsUnique using my def; else don't unfold it.\nAlso, if first = true, then unfold constants in list dontUnfold; otherwise don't.\nIf rep = true, unfold repeatedly.\nLet whnfCore handle everything except unfolding of constants.\nDo all normalization up to first unfolding of a definition; on next call do that unfolding\n-/\n\ndef fixElt (e : Expr) (doFix : Bool) : TacticM Expr := do\n if doFix then\n match e with --if e is \"set elt\", change to \"elt \u2208 set\"\n | app st elt =>\n let tp \u2190 Meta.inferType st\n match tp with\n | app (const ``Set [lev]) t =>\n return (mkApp5 (mkConst ``Membership.mem [lev, lev])\n t\n (mkApp (mkConst ``Set [lev]) t)\n (mkApp (mkConst ``Set.instMembershipSet [lev]) t)\n elt\n st)\n | _ => return e\n | _ => return e\n else\n return e\n\n--Unfold if possible, else return input. Input should have no MData.\npartial def unfoldHeadCore (e : Expr) (first rep : Bool) : TacticM Expr := do\n match e with --Don't unfold {x | p} if not applied to anything\n | (app (app (const ``setOf _) _) _) => return e\n | _ => pure ()\n let (h, args) := getHeadData e\n -- First let e1 = result of one unfolding, or handle negation, or return e\n let e1 \u2190 match h with\n | const c levs =>\n match (c, levs, args) with\n | (``Not, _, [l]) => return mkNot (\u2190 unfoldHeadCore (consumeMData l) first rep)\n | (``ExistsUnique, [lev], [r, l]) =>\n if first then\n pure (applyToExData unfoldExUn lev l r)\n else\n return e\n | _ =>\n if !first then\n if c \u2208 dontUnfold then\n return e\n if c \u2208 dontUnfoldNum then\n match args[3]! with\n | const nc _ => if nc \u2208 numNames then return e\n | _ => pure ()\n let edo \u2190 Meta.unfoldDefinition? e\n match edo with\n | some ed => pure ed\n | none => return (\u2190 fixElt e rep)\n | _ =>\n let ew \u2190 Meta.whnfCore e\n if ew == e then\n return (\u2190 fixElt e rep)\n else\n pure ew\n if rep then\n return (\u2190 unfoldHeadCore (consumeMData e1) false true)\n else\n return e1\n\ndef unfoldHead (e : Expr) (tac : Name) (first rep : Bool) : TacticM Expr := do\n let e1 := consumeMData e\n let e2 \u2190 unfoldHeadCore e1 first rep\n if e2 == e1 then\n myFail tac \"failed to unfold definition\"\n else\n return e2\n\n-- whnf, but don't unfold ``ExistsUnique\ndef whnfNotExUn (e : Expr) : TacticM Expr :=\n Meta.whnfHeadPred e (fun x => return !(x.isAppOf ``ExistsUnique))\n\n-- w = 0 : no whnf, w = 1 : whnfNotExun, w = 2 : full whnf\ndef exprFromPf (t : Term) (w : Nat) : TacticM Expr := do\n let p \u2190 elabTerm t none\n let e \u2190 instantiateMVars (\u2190 Meta.inferType p)\n match w with\n | 0 => return e\n | 1 => whnfNotExUn e\n | _ => Meta.whnf e\n\n--Add new hypothesis with name n, asserting form, proven by pfstx.\ndef doHave (n : Name) (form : Expr) (pfstx : Syntax) : TacticM Unit := do\n let goal \u2190 getMainGoal\n let oldtar \u2190 getType goal\n let pf \u2190 elabTermEnsuringType pfstx form\n let mvarIdNew \u2190 assert goal n form pf\n let (_, newGoal) \u2190 intro1P mvarIdNew --blank is FVarId of new hyp.\n let newtar \u2190 getType newGoal\n if (oldtar != newtar) && (\u2190 Meta.isExprDefEq oldtar newtar) then\n --intro1P sometimes changes target to something def. equal. Put it back to original\n replaceMainGoal [\u2190 newGoal.replaceTargetDefEq oldtar]\n else\n replaceMainGoal [newGoal]\n\n--Set goal to be form; pfstx is proof that it suffices.\ndef doSuffices (form : Expr) (pfstx : Syntax) : TacticM Unit := do\n let goal \u2190 getMainGoal\n let tag \u2190 getTag goal\n let target \u2190 getType goal\n let imp \u2190 mkArrow form target\n let pf \u2190 elabTermEnsuringType pfstx imp\n let newTarget \u2190 Meta.mkFreshExprSyntheticOpaqueMVar form tag\n assign goal (mkApp pf newTarget)\n replaceMainGoal [newTarget.mvarId!]\n\n--Do rewrite; symm says whether to reverse direction, rule is Term for rule, l is optional location\ndef doRewrite (symm : Bool) (rule : Term) (l : Option OneLoc) : TacticM Unit := do\n match l with\n | some id =>\n let idstx : Ident := \u27e8id.raw[1]\u27e9\n if symm then\n evalTactic (\u2190 `(tactic| rewrite [\u2190 $rule:term] at $idstx:ident))\n else\n evalTactic (\u2190 `(tactic| rewrite [$rule:term] at $idstx:ident))\n | none =>\n if symm then\n evalTactic (\u2190 `(tactic| rewrite [\u2190 $rule:term]))\n else\n evalTactic (\u2190 `(tactic| rewrite [$rule:term]))\n\n--Swap first two goals, if there are at least two\ndef doSwap : TacticM Unit := do\n let g \u2190 getGoals\n let ng := match g with\n | g1 :: (g2 :: rest) => g2 :: (g1 :: rest)\n | _ => g\n setGoals ng\n\n/- Functions for all equivalence tactics: contrapos, demorgan, quant_neg, conditional, double_neg -/\ndef ruleType := Name \u00d7 Expr\n\ndef equivMakeRule (f : Expr)\n (ruleFunc : Expr \u2192 TacticM ruleType) : TacticM ruleType := do\n let (rule, res) \u2190 ruleFunc f\n return (rule, mkIff f res)\n\ndef equivRuleFromForm (p : Expr)\n (ruleFunc : Expr \u2192 TacticM ruleType) : TacticM ruleType := do\n try\n equivMakeRule p ruleFunc\n catch ex =>\n match (\u2190 getPropForm p) with\n | PropForm.iff l r =>\n try\n equivMakeRule l ruleFunc\n catch _ =>\n equivMakeRule r ruleFunc\n | _ => throw ex\n\ndef equivRule (f : Option ColonTerm) (l : Option OneLoc)\n (ruleFunc : Expr \u2192 TacticM ruleType) : TacticM ruleType := do\n match f with\n | some fs => equivMakeRule (\u2190 elabTerm fs.raw[1] none) ruleFunc\n | none => equivRuleFromForm (\u2190 formFromLoc l) ruleFunc\n\ndef doReplace (tac : Name) (l : Option OneLoc) (res : Expr) (pf : Syntax) : TacticM Unit := do\n let hn \u2190 mkFreshUserName `h\n doHave hn res pf\n let h := mkIdent hn\n let ht : Term := \u27e8h.raw\u27e9\n try\n doRewrite false ht l\n evalTactic (\u2190 `(tactic| clear $h:ident)) -- Could also do: (try apply Iff.refl); try assumption\n catch _ =>\n evalTactic (\u2190 `(tactic| clear $h:ident))\n myFail tac \"target expression not found\"\n\ndef doEquivTac (f : Option ColonTerm) (l : Option OneLoc)\n (tac : Name) (ruleFunc : Expr \u2192 TacticM ruleType) : TacticM Unit :=\n withMainContext do\n let (rule, res) \u2190 equivRule f l ruleFunc\n doReplace tac l res (mkIdent rule)\n\n/- contrapos tactic -/\ndef cpRule (form : Expr) : TacticM ruleType := do\n match (\u2190 getPropForm form) with\n | PropForm.implies l r => match (\u2190 getPropForm l) with\n | PropForm.not nl => match (\u2190 getPropForm r) with\n | PropForm.not nr =>\n return (`not_imp_not, (\u2190 mkArrow nr nl))\n | _ =>\n return (`not_imp_comm, (\u2190 mkArrow (mkNot r) nl))\n | _ => match (\u2190 getPropForm r) with\n | PropForm.not nr =>\n return (`imp_not_comm, (\u2190 mkArrow nr (mkNot l)))\n | _ =>\n return (`not_imp_not.symm, (\u2190 mkArrow (mkNot r) (mkNot l)))\n | _ => myFail `contrapos \"contrapositive law doesn't apply\"\n\nelab \"contrapos\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `contrapos cpRule\n\n/- demorgan tactic -/\ndef dmRuleFromInfoNoNeg (l r : Expr) (conn : Expr \u2192 Expr \u2192 Expr) (rs : Array Name) : TacticM ruleType := do\n match (\u2190 getPropForm l) with\n | PropForm.not nl =>\n match (\u2190 getPropForm r) with\n | PropForm.not nr => return (rs[0]!, conn nl nr)\n | _ => return (rs[1]!, conn nl (mkNot r))\n | _ =>\n match (\u2190 getPropForm r) with\n | PropForm.not nr => return (rs[2]!, conn (mkNot l) nr)\n | _ => return (rs[3]!, conn (mkNot l) (mkNot r))\n\ndef dmRuleFromInfo (l r : Expr) (conn : Expr \u2192 Expr \u2192 Expr) (n : Bool) (rs : Array Name) : TacticM ruleType := do\n let p \u2190 dmRuleFromInfoNoNeg l r conn rs\n if n then\n return (p.1, mkNot p.2)\n else\n return p\n\ndef dmRule (form : Expr) : TacticM ruleType := do\n match (\u2190 getPropForm form) with\n | PropForm.not a => match (\u2190 getPropForm a) with\n | PropForm.and l r =>\n dmRuleFromInfo l r mkOr false\n #[`or_iff_not_and_not.symm, `not_not_and_distrib, `not_and_not_distrib, `not_and_or]\n | PropForm.or l r =>\n dmRuleFromInfo l r mkAnd false\n #[`and_iff_not_or_not.symm, `not_not_or_distrib, `not_or_not_distrib, `not_or]\n | _ => myFail `demorgan \"De Morgan's laws don't apply\"\n | PropForm.and l r =>\n dmRuleFromInfo l r mkOr true\n #[`not_or.symm, `not_or_not_distrib.symm, `not_not_or_distrib.symm, `and_iff_not_or_not]\n | PropForm.or l r =>\n dmRuleFromInfo l r mkAnd true\n #[`not_and_or.symm, `not_and_not_distrib.symm, `not_not_and_distrib.symm, `or_iff_not_and_not]\n | _ => myFail `demorgan \"De Morgan's laws don't apply\"\n\nelab \"demorgan\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `demorgan dmRule\n\n/- quant_neg tactic -/\ndef qnRuleFromInfoNoNeg (v : Name) (t b : Expr) (qf : Name \u2192 BinderInfo \u2192 Expr \u2192 Expr \u2192 Expr)\n (rs : Name \u00d7 Name) : TacticM ruleType := do\n let f := mkLambda `x BinderInfo.default t b\n let negres \u2190 Meta.lambdaTelescope f fun fvs e => do\n match (\u2190 getPropForm e) with\n | PropForm.not ne => return some (\u2190 Meta.mkLambdaFVars fvs ne)\n | _ => return none\n match negres with\n | some ne => match ne with\n | lam _ _ nb _ => return (rs.1, qf v BinderInfo.default t nb)\n | _ => return (rs.2, qf v BinderInfo.default t (mkNot b))\n | none => return (rs.2, qf v BinderInfo.default t (mkNot b))\n\ndef qnRuleFromInfo (v : Name) (t b : Expr) (qf : Name \u2192 BinderInfo \u2192 Expr \u2192 Expr \u2192 Expr)\n (n : Bool) (rs : Name \u00d7 Name) : TacticM ruleType := do\n let p \u2190 qnRuleFromInfoNoNeg v t b qf rs\n if n then\n return (p.1, mkNot p.2)\n else\n return p\n\ndef qnRule (form : Expr) : TacticM ruleType := do\n match (\u2190 getPropForm form) with\n | PropForm.not p => match (\u2190 getPropForm p) with\n | PropForm.all v t b _ =>\n qnRuleFromInfo v t b (mkExists (\u2190 Meta.getLevel t)) false\n (`not_forall_not, `not_forall)\n | PropForm.ex _ v t b _ =>\n qnRuleFromInfo v t b mkForall false\n (`not_exists_not, `not_exists)\n | _ => myFail `quant_neg \"quantifier negation laws don't apply\"\n | PropForm.all v t b _ =>\n qnRuleFromInfo v t b (mkExists (\u2190 Meta.getLevel t)) true\n (`not_exists.symm, `not_exists_not.symm)\n | PropForm.ex _ v t b _ =>\n qnRuleFromInfo v t b mkForall true\n (`not_forall.symm, `not_forall_not.symm)\n | _ => myFail `quant_neg \"quantifier negation laws don't apply\"\n\nelab \"quant_neg\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `quant_neg qnRule\n\n/- conditional tactic -/\ndef cdlRule (form : Expr) : TacticM ruleType := do\n match (\u2190 getPropForm form) with\n | PropForm.not p => match (\u2190 getPropForm p) with\n | PropForm.implies l r => match (\u2190 getPropForm r) with\n | PropForm.not nr => return (`not_imp_not_iff_and, mkAnd l nr)\n | _ => return (`Classical.not_imp, mkAnd l (mkNot r))\n | _ => myFail `conditional \"conditional laws don't apply\"\n | PropForm.implies l r => match (\u2190 getPropForm l) with\n | PropForm.not nl => return (`or_iff_not_imp_left.symm, mkOr nl r)\n | _ => return (`imp_iff_not_or, mkOr (mkNot l) r)\n | PropForm.and l r => match (\u2190 getPropForm r) with\n | PropForm.not nr => return (`Classical.not_imp.symm, mkNot (\u2190 mkArrow l nr))\n | _ => match (\u2190 getPropForm l) with\n | PropForm.not nl => return (`not_imp_iff_not_and.symm, mkNot (\u2190 mkArrow r nl))\n | _ => return (`not_imp_not_iff_and.symm, mkNot (\u2190 mkArrow l (mkNot r)))\n | PropForm.or l r => match (\u2190 getPropForm l) with\n | PropForm.not nl => return (`imp_iff_not_or.symm, (\u2190 mkArrow nl r))\n | _ => match (\u2190 getPropForm r) with\n | PropForm.not nr => return (`imp_iff_or_not.symm, (\u2190 mkArrow nr l))\n | _ => return (`or_iff_not_imp_left, (\u2190 mkArrow (mkNot l) r))\n | _ => myFail `conditional \"conditional laws don't apply\"\n\nelab \"conditional\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `conditional cdlRule\n\n/- double_neg tactic -/\ndef dnRule (form : Expr) : TacticM ruleType := do\n match (\u2190 getPropForm form) with\n | PropForm.not p1 => match (\u2190 getPropForm p1) with\n | PropForm.not p2 => return (`Classical.not_not, p2)\n | _ => myFail `double_neg \"double negation law doesn't apply\"\n | _ => myFail `double_neg \"double negation law doesn't apply\"\n\nelab \"double_neg\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `double_neg dnRule\n\n/- bicond_neg tactic\nNote converts P \u2194 Q to \u00ac(\u00acP \u2194 Q).\nSo to convert only one side of \u2194, must use : [term to convert] -/\ndef binegRule (form : Expr) : TacticM ruleType := do\n match (\u2190 getPropForm form) with\n | PropForm.not p => match (\u2190 getPropForm p) with\n | PropForm.iff l r => match (\u2190 getPropForm l) with\n | PropForm.not nl => return (`not_not_iff, mkIff nl r)\n | _ => return (`not_iff, mkIff (mkNot l) r)\n | _ => myFail `bicond_neg \"biconditional negation law doesn't apply\"\n | PropForm.iff l r => match (\u2190 getPropForm l) with\n | PropForm.not nl => return (`not_iff.symm, mkNot (mkIff nl r))\n | _ => return (`not_not_iff.symm, mkNot (mkIff (mkNot l) r))\n | _ => myFail `bicond_neg \"biconditional negation law doesn't apply\"\n\nelab \"bicond_neg\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `bicond_neg binegRule\n\n-- Give error if any ident in i is already in use. Is this right thing to do in all cases?\npartial def checkIdUsed (tac : Name) (i : Syntax) : TacticM Unit := do\n match i with\n | .missing => return ()\n | .node _ _ as => for a in as do checkIdUsed tac a\n | .atom _ _ => return ()\n | .ident _ _ v _ =>\n if (\u2190 getLCtx).usesUserName v then\n myFail tac (\"identifier \" ++ (toString v) ++ \" already in use\")\n else\n return ()\n\n-- Get label from \"with\" clause, or default label. Used by several tactics\ndef getLabel (tac : Name) (w : Option WithId) (dflt : Ident := mkIdent `this) : TacticM Ident := do\n match w with\n | some h =>\n let i := h.raw[1]\n checkIdUsed tac i\n return \u27e8i\u27e9\n | none => return dflt\n\ndef isLocalVar (s : Syntax) : TacticM Bool := do\n match s with\n | .ident _ _ v _ => return (\u2190 getLCtx).usesUserName v\n | _ => return False\n\n/- or_left and or_right tactics -/\ndef negData (e : Expr) : TacticM (Expr \u00d7 Bool) := do\n match (\u2190 getPropForm e) with\n | PropForm.not ne => return (ne, true)\n | _ => return (e, false)\n\ndef orstrat (tac : Name) (w : Option WithId) (left : Bool) : TacticM Unit :=\n withMainContext do\n let label \u2190 getLabel tac w\n let d \u2190 getMainDecl\n let t \u2190 Meta.whnf (\u2190 instantiateMVars d.type)\n match (\u2190 getPropForm t) with\n | PropForm.or l r => do\n let (form, neg) \u2190 negData (if left then r else l)\n let goalName := d.userName\n let emn \u2190 mkFreshUserName `h\n let emi := mkIdent emn\n doHave emn (mkOr form (mkNot form)) (\u2190 `(em _))\n evalTactic (\u2190 `(tactic|refine Or.elim $emi:ident ?_ ?_))\n if neg then doSwap\n let (rule1, rule2) :=\n if left then\n (mkIdent ``Or.inr, mkIdent ``Or.inl)\n else\n (mkIdent ``Or.inl, mkIdent ``Or.inr)\n evalTactic (\u2190 `(tactic| exact fun x => $rule1:ident x))\n evalTactic (\u2190 `(tactic| intro $label:ident; refine $rule2:ident ?_; clear $emi:ident))\n let newGoal \u2190 getMainGoal\n setUserName newGoal goalName\n | _ => myFail tac \"goal is not a disjunction\"\n\nelab \"or_left\" w:(withId)? : tactic => orstrat `or_left w true\nelab \"or_right\" w:(withId)? : tactic => orstrat `or_right w false\n\n/- disj_syll tactic -/\ndef matchFirstNeg (e1 e2 : Expr) : TacticM Bool := do\n match (\u2190 getPropForm e1) with\n | PropForm.not ne1 => Meta.isExprDefEq ne1 e2\n | _ => return false\n\n--1st coord: does one match neg of other? 2nd coord: does first match neg of second?\ndef matchNeg (e1 e2 : Expr) : TacticM (Bool \u00d7 Bool) := do\n if (\u2190 matchFirstNeg e1 e2) then\n return (true, true)\n else\n return ((\u2190 matchFirstNeg e2 e1), false)\n\n--1st coord: Does neg contradict right side of disj? (else left side)\n--2nd coord: Is disjunct negation of neg? (else neg is negation of disj)\ndef DisjSyllData (disj neg : Expr) : TacticM (Bool \u00d7 Bool) := do\n match (\u2190 getPropForm disj) with\n | PropForm.or l r =>\n let (isneg, disjneg) \u2190 matchNeg l neg\n if isneg then\n return (false, disjneg)\n else\n let (isneg, disjneg) \u2190 matchNeg r neg\n if isneg then\n return (true, disjneg)\n else\n myFail `disj_syll \"disjunctive syllogism rule doesn't apply\"\n | _ => myFail `disj_syll \"disjunctive syllogism rule doesn't apply\"\n\ndef parseIdOrTerm (it : IdOrTerm) : Term :=\n let s := it.raw[0]\n match s with\n | .ident .. => \u27e8s\u27e9\n | _ => \u27e8s[1]\u27e9\n\nelab \"disj_syll\" dIOrT:idOrTerm nIOrT:idOrTerm w:(withId)? : tactic =>\n withMainContext do\n let d := parseIdOrTerm dIOrT\n let n := parseIdOrTerm nIOrT\n let disj \u2190 exprFromPf d 2\n let neg \u2190 exprFromPf n 0\n let (dId, deflabel) :=\n if (\u2190 isLocalVar d.raw) then\n (true, \u27e8d.raw\u27e9)\n else\n (false, mkIdent `this)\n let label \u2190 getLabel `disj_syll w deflabel\n let (conright, disjneg) \u2190 DisjSyllData disj neg\n let goalName := (\u2190 getMainDecl).userName\n evalTactic (\u2190 `(tactic| refine Or.elim $d ?_ ?_))\n if conright then doSwap\n if disjneg then\n evalTactic (\u2190 `(tactic| exact fun x => absurd $n x))\n else\n evalTactic (\u2190 `(tactic| exact fun x => absurd x $n))\n if (dId && (w == none)) then evalTactic (\u2190 `(tactic| clear $label:ident))\n evalTactic (\u2190 `(tactic| intro $label:ident))\n let newGoal \u2190 getMainGoal\n setUserName newGoal goalName\n\n/- contradict tactic -/\ndef ensureContra (w : Option WithId) : TacticM Unit :=\n withMainContext do\n let label \u2190 getLabel `contradict w\n let t \u2190 getMainTarget\n match (\u2190 getPropForm t) with\n | PropForm.f => return ()\n | _ => evalTactic (\u2190 `(tactic| by_contra $label:ident))\n\nelab \"contradict\" h:term w:(withId)? : tactic => do\n ensureContra w\n withMainContext do\n --let tocon \u2190 formFromIdent h.raw\n let tocon \u2190 exprFromPf h 0\n match (\u2190 getPropForm tocon) with\n | PropForm.not p =>\n doSuffices p (\u2190 `(fun x => $h x))\n | _ =>\n doSuffices (mkNot tocon) (\u2190 `(fun x => x $h))\n\n/- define, def_step, and whnf tactics -/\ndef unfoldOrWhnf (tac: Name) (e : Expr) (w rep : Bool) : TacticM Expr := do\n if w then\n match (\u2190 getPropForm e) with\n | PropForm.exun lev v t b bi => return unfoldExUn lev v t b bi\n | _ => whnfNotExUn e\n else\n unfoldHead e tac true rep\n\ndef doDefine (tac : Name) (f : Option ColonTerm) (l : Option OneLoc) (w rep : Bool) : TacticM Unit :=\n withMainContext do\n let e \u2190 match f with\n | some fs => elabTerm fs.raw[1] none\n | none => formFromLoc l\n let e' \u2190 unfoldOrWhnf tac e w rep\n doReplace tac l (\u2190 Meta.mkEq e e') (\u2190 `(Eq.refl _))\n\nelab \"define\" f:(colonTerm)? l:(oneLoc)? : tactic => doDefine `define f l false true\nelab \"whnf\" f:(colonTerm)? l:(oneLoc)? : tactic => doDefine `whnf f l true true\nelab \"def_step\" f:(colonTerm)? l:(oneLoc)? : tactic => doDefine `def_step f l false false\n\n/- definition and definition! tactics -/\n--Context set in doDefinition, which calls these functions\ndef getDefineFormLabel (f : Option ColonTerm) (l : Option OneLoc) : TacticM (Expr \u00d7 Name) := do\n match f with\n | some t => return (\u2190 elabTerm t.raw[1] none, `this)\n | none => match l with\n | some h => do\n let hs := h.raw[1]\n return (\u2190 formFromIdent hs, Name.mkStr hs.getId \"def\")\n | none => return (\u2190 getMainTarget, `goal.def)\n\n-- use Iff for propositions, = for other types\ndef mkRel (e1 e2 : Expr) (prop : Bool) : TacticM Expr :=\n if prop then\n return mkIff e1 e2\n else\n Meta.mkEq e1 e2\n\n-- repeatedly assert definition equivalences or equations, numbering steps\npartial def doDefinitionRep (label : Name) (e e1 : Expr) (prop : Bool) (rule : Ident) (firstNum : Nat) : TacticM Unit := do\n let e' \u2190 unfoldHead e1 `definition (firstNum == 1) false\n let res \u2190 mkRel e e' prop\n doHave (Name.appendIndexAfter label firstNum) res (\u2190 `($rule _))\n try\n withMainContext (doDefinitionRep label e e' prop rule (firstNum + 1)) -- Context changes each time through\n catch _ =>\n return ()\n\ndef doDefinition (all : Bool) (f : Option ColonTerm) (l : Option OneLoc) (wid : Option WithId) : TacticM Unit :=\n withMainContext do\n let (e, deflabel) \u2190 getDefineFormLabel f l\n let label \u2190 getLabel `definition wid (mkIdent deflabel)\n let labeln := label.getId\n let (prop, rule) := if (\u2190 exprIsProp e) then\n (true, mkIdent ``Iff.refl)\n else\n (false, mkIdent ``Eq.refl)\n if all then\n doDefinitionRep labeln e e prop rule 1\n else\n let e' \u2190 unfoldHead e `definition true true\n let res \u2190 mkRel e e' prop\n doHave labeln res (\u2190 `($rule _))\n\nelab \"definition\" f:(colonTerm) wid:(withId)? : tactic => doDefinition false (some f) none wid\nelab \"definition\" l:(oneLoc)? wid:(withId)? : tactic => doDefinition false none l wid\nelab \"definition!\" f:(colonTerm) wid:(withId)? : tactic => doDefinition true (some f) none wid\nelab \"definition!\" l:(oneLoc)? wid:(withId)? : tactic => doDefinition true none l wid\n\ndef addToName (n : Name) (s : String) : Name :=\n Name.modifyBase n (fun x => Name.mkStr x s)\n\n--Bool is whether or not to clear \"or\" given; Idents for two cases\ndef setUpCases (t : Term) (wids : Option With2Ids) : TacticM (Bool \u00d7 Ident \u00d7 Ident) := do\n match wids with\n | some ids =>\n let id1s := ids.raw[1]\n checkIdUsed `by_cases id1s\n let id1 : Ident := \u27e8id1s\u27e9\n match ids.raw[2].getArgs[1]? with\n | some id2 =>\n checkIdUsed `by_cases id2\n return (false, id1, \u27e8id2\u27e9)\n | none => return (false, id1, id1)\n | none =>\n if (\u2190 isLocalVar t.raw) then\n let tid : Ident := \u27e8t.raw\u27e9\n return (true, tid, tid)\n else\n let thisId := mkIdent `this\n return (false, thisId, thisId)\n\ndef fixCase (clear : Bool) (label : Ident) (g : Name) (c : String) : TacticM Unit := do\n if clear then\n evalTactic (\u2190 `(tactic| clear $label))\n evalTactic (\u2190 `(tactic| intro $label:ident))\n setUserName (\u2190 getMainGoal) (addToName g c)\n doSwap\n\nelab \"by_cases\" \"on\" t:term wids:(with2Ids)? : tactic =>\n withMainContext do\n let e \u2190 exprFromPf t 2\n match (\u2190 getPropForm e) with\n | PropForm.or _ _ =>\n let (clear, label1, label2) \u2190 setUpCases t wids\n let goalname := (\u2190 getMainDecl).userName\n evalTactic (\u2190 `(tactic| refine Or.elim $t ?_ ?_))\n fixCase clear label1 goalname \"Case_1\"\n fixCase clear label2 goalname \"Case_2\"\n | _ => myFail `by_cases \"hypothesis is not a disjunction\"\n\n/- exists_unique tactic -/\ndef mkUn (lev: Level) (v : Name) (t b : Expr) : TacticM Expr := do\n let v1 := Name.appendIndexAfter v 1\n let v2 := Name.appendIndexAfter v 2\n let f1 := mkLambda v1 BinderInfo.default t b\n let f2 := mkLambda v2 BinderInfo.default t b\n Meta.lambdaTelescope f1 fun fv1 e1 =>\n Meta.lambdaTelescope f2 fun fv2 e2 => do\n let body \u2190 mkArrow e1 (\u2190 mkArrow e2\n (mkApp3 (const ``Eq [lev]) t fv1[0]! fv2[0]!))\n Meta.mkForallFVars (fv1.push fv2[0]!) body\n\nelab \"exists_unique\" : tactic => do\n let goal \u2190 getMainGoal\n withContext goal do\n let d \u2190 getDecl goal\n let goalname := d.userName\n let tar \u2190 instantiateMVars d.type\n match (\u2190 getPropForm tar) with\n | PropForm.exun lev v t b _ =>\n let un \u2190 mkUn lev v t b\n let ex := mkExists lev v BinderInfo.default t b\n let h \u2190 mkFreshUserName `h\n let hid := mkIdent h\n let hex := (mkForall `a BinderInfo.default ex\n (mkForall `b BinderInfo.default un tar))\n doHave h hex (\u2190 `(exists_unique_of_exists_of_unique))\n evalTactic (\u2190 `(tactic| refine $hid ?_ ?_; clear $hid))\n setUserName (\u2190 getMainGoal) (addToName goalname \"Existence\")\n doSwap\n evalTactic (\u2190 `(tactic| clear $hid))\n setUserName (\u2190 getMainGoal) (addToName goalname \"Uniqueness\")\n doSwap\n | _ => myFail `exists_unique \"goal is not a unique existence statement\"\n\n/- obtain tactic -/\ndef parseIdOrTerm?Type (tac : Name) (it : IdOrTerm?Type) : TacticM (Term \u00d7 (Option Term)) := do\n let s := it.raw[0]\n let res := match s with\n | .ident .. => (\u27e8s\u27e9, none)\n | _ => match s[2].getArgs[1]? with\n | some t => (\u27e8s[1]\u27e9, some \u27e8t\u27e9)\n | none => (\u27e8s[1]\u27e9, none)\n checkIdUsed tac res.1.raw\n return res\n\n/- Old version\ndef doIntroOption (i : Term) (t : Option Term) : TacticM Unit := do\n match t with\n | some tt => --evalTactic (\u2190 `(tactic| intro ($i : $tt)))\n --Above fails in Chap8Part1 line 572. Problem seems to be that\n --{a} is not understood as a Set despite default instance declarations above.\n evalTactic (\u2190 `(tactic| intro h; match @h with | ($i : $tt) => ?_; try clear h))\n | none => evalTactic (\u2190 `(tactic| intro $i:term))\n-/\n\n--Imitating mkHasTypeButIsExpectedMsg\ndef mkDeclaredTypeButIsExpectedMsg (decType expectedType : Expr) : MetaM MessageData := do\n try\n let decTypeType \u2190 Meta.inferType decType\n let expectedTypeType \u2190 Meta.inferType expectedType\n let (decType, expectedType) \u2190 Meta.addPPExplicitToExposeDiff decType expectedType\n let (decTypeType, expectedTypeType) \u2190 Meta.addPPExplicitToExposeDiff decTypeType expectedTypeType\n return m!\"is declared to have type{indentD m!\"{decType} : {decTypeType}\"}\\nbut is expected to have type{indentD m!\"{expectedType} : {expectedTypeType}\"}\"\n catch _ =>\n let (decType, expectedType) \u2190 Meta.addPPExplicitToExposeDiff decType expectedType\n return m!\"is declared to have type{indentExpr decType}\\nbut is expected to have type{indentExpr expectedType}\"\n\ndef doIntroOption (tac : Name) (i : Term) (t : Option Term) : TacticM Unit := withMainContext do\n match t with\n | some tt =>\n let et \u2190 elabTerm tt none\n let goal \u2190 getMainGoal\n let h \u2190 mkFreshUserName `h\n let hid := mkIdent h\n let (fvid, goal2) \u2190 goal.intro h\n replaceMainGoal [goal2]\n withMainContext do\n let fv := mkFVar fvid\n let fvt \u2190 Meta.inferType fv\n if (\u2190 Meta.isDefEq et fvt) then\n replaceMainGoal [\u2190 goal2.replaceLocalDeclDefEq fvid et]\n evalTactic (\u2190 `(tactic| match @$hid with | ($i : _) => ?_; try clear $hid))\n else\n Meta.throwTacticEx tac goal m!\"type mismatch: {i} {\u2190 mkDeclaredTypeButIsExpectedMsg et fvt}\"\n | none => evalTactic (\u2190 `(tactic| intro $i:term))\n\ndef doObtain (itw ith : IdOrTerm?Type) (tm : Term) : TacticM Unit :=\n withMainContext do\n let e \u2190 exprFromPf tm 1\n match (\u2190 getPropForm e) with\n | PropForm.ex _ _ _ _ _ =>\n let (wi, wt) \u2190 parseIdOrTerm?Type `obtain itw\n let (hi, ht) \u2190 parseIdOrTerm?Type `obtain ith\n evalTactic (\u2190 `(tactic| refine Exists.elim $tm ?_))\n doIntroOption `obtain wi wt\n doIntroOption `obtain hi ht\n | _ => myFail `obtain \"hypothesis is not an existence statement\"\n\ntheorem exun_elim {\u03b1 : Sort u} {p : \u03b1 \u2192 Prop} {b : Prop}\n (h2 : \u2203! x, p x) (h1 : \u2200 x, p x \u2192 (\u2200 y z, p y \u2192 p z \u2192 y = z) \u2192 b) : b := by\n apply ExistsUnique.elim h2\n intro x h3 h4\n apply h1 x h3\n intro y z h5 h6\n have h7 := h4 y h5\n have h8 := h4 z h6\n rw [h7,h8]\n\ndef doObtainExUn (itw ith1 ith2 : IdOrTerm?Type) (tm : Term) : TacticM Unit :=\n withMainContext do\n let e \u2190 exprFromPf tm 1\n match (\u2190 getPropForm e) with\n | PropForm.exun lev v t b _ =>\n let (wi, wt) \u2190 parseIdOrTerm?Type `obtain itw\n let (h1i, h1t) \u2190 parseIdOrTerm?Type `obtain ith1\n let (h2i, h2t) \u2190 parseIdOrTerm?Type `obtain ith2\n let tar \u2190 getMainTarget\n let un \u2190 mkUn lev v t b\n let exun := mkForall v BinderInfo.default t (\u2190 mkArrow b (\u2190 mkArrow un tar))\n let h \u2190 mkFreshUserName `h\n let hid := mkIdent h\n doHave h (\u2190 mkArrow exun tar) (\u2190 `(exun_elim $tm))\n evalTactic (\u2190 `(tactic| refine $hid ?_; clear $hid))\n doIntroOption `obtain wi wt\n doIntroOption `obtain h1i h1t\n doIntroOption `obtain h2i h2t\n | _ => myFail `obtain \"hypothesis is not a unique existence statement\"\n\n--Make 1 assertion for existential, 2 for unique existential\nelab \"obtain\" itw:idOrTerm?Type ith:idOrTerm?Type \" from \" t:term : tactic =>\n doObtain itw ith t\nelab \"obtain\" itw:idOrTerm?Type ith1:idOrTerm?Type ith2:idOrTerm?Type \" from \" t:term : tactic =>\n doObtainExUn itw ith1 ith2 t\n\n/- assume and fix tactics -/\ndef doAssume (w : Term) (t : Option Term) : TacticM Unit :=\n withMainContext do\n checkIdUsed `assume w\n match (\u2190 getPropForm (\u2190 Meta.whnf (\u2190 getMainTarget))) with\n | PropForm.implies _ _ => doIntroOption `assume w t\n --| PropForm.not _ => doIntroOption w t --Not necessary--whnf will have changed to implies\n | _ => myFail `assume \"goal is not a conditional statement\"\n\ndef doFix (w : Term) (t : Option Term) : TacticM Unit :=\n withMainContext do\n checkIdUsed `fix w\n match (\u2190 getPropForm (\u2190 Meta.whnf (\u2190 getMainTarget))) with\n | PropForm.all _ _ _ _ => doIntroOption `fix w t\n | _ => myFail `fix \"goal is not a universally quantified statement\"\n\nelab \"assume\" w:term : tactic => doAssume w none\nelab \"assume\" w:term \" : \" t:term : tactic => doAssume w (some t)\nelab \"fix\" w:term : tactic => doFix w none\nelab \"fix\" w:term \" : \" t:term : tactic => doFix w (some t)\n\n/- show tactic: allow either \"from\" or \":=\" Probably best to stick to \"from\" -/\nmacro \"show \" c:term \" from \" p:term : tactic => `(tactic| {show $c; exact $p})\nmacro \"show \" c:term \" := \" p:term : tactic => `(tactic| {show $c; exact $p})\nmacro \"show \" c:term \" by \" p:tactic : tactic => `(tactic| {show $c; $p})\n\n", "theoremStatement": "theorem induc_from (P : Nat \u2192 Prop) (k : Nat) (h1 : P k) (h2 : (\u2200 n \u2265 k, P n \u2192 P (n+1))) :\n \u2200 n \u2265 k, P n ", "theoremName": "HTPI.induc_from", "fileCreated": {"commit": "94654f92e3e5e5cd0ba367d89e9ba3f1ed7c911e", "date": "2022-09-27"}, "theoremCreated": {"commit": "1892a424703ed78fdba66ef6f1d72bab93c740ea", "date": "2023-01-20"}, "file": "htpi/HTPILib/HTPIDefs.lean", "module": "HTPILib.HTPIDefs", "jsonFile": "HTPILib.HTPIDefs.jsonl", "positionMetadata": {"lineInFile": 982, "tokenPositionInFile": 39003, "theoremPositionInFile": 128}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 26, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", 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"Std.Data.String.Basic", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply @Nat.rec\n assume h3\n have h4 : k = 0 := Nat.eq_zero_of_le_zero h3\n rewrite [h4] at h1\n exact h1\n fix n\n assume h3\n assume h4\n have h5 : k < n + 1 \u2228 k = n + 1 := LE.le.lt_or_eq_dec h4\n by_cases on h5\n have h6 : k \u2264 n := Nat.le_of_lt_succ h5\n have h7 := h3 h6\n exact h2 n h6 h7\n rewrite [h5] at h1\n exact h1", "proofType": "tactic", "proofLengthLines": 15, "proofLengthTokens": 333}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\n", "theoremStatement": "lemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 ", "theoremName": "HTPI.list_nil_iff_prod_one", "fileCreated": {"commit": 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"Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done", "proofType": "tactic", "proofLengthLines": 17, "proofLengthTokens": 425}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\nlemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) :=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)\n\nlemma G_perm_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n perm_below m (G m a) := And.intro (G_maps_below m a)\n (And.intro (G_one_one_below h1) (G_onto_below h1))\n\n--Permuting a product of congruence classes doesn't change product\nlemma swap_fst (u v : Nat) : swap u v u = v := by\n define : swap u v u\n --Goal : (if u = u then v else if u = v then u else u) = v\n have h : u = u := by rfl\n rewrite [if_pos h]\n rfl\n done\n\nlemma swap_snd (u v : Nat) : swap u v v = u := by\n define : swap u v v\n by_cases h1 : v = u\n \u00b7 -- Case 1. h1 : v = u\n rewrite [if_pos h1]\n show v = u from h1\n done\n \u00b7 -- Case 2. h1 : v \u2260 u\n rewrite [if_neg h1]\n have h2 : v = v := by rfl\n rewrite [if_pos h2]\n rfl\n done\n done\n\nlemma swap_other {u v i : Nat} (h1 : i \u2260 u) (h2 : i \u2260 v) : swap u v i = i := by\n define : swap u v i\n rewrite [if_neg h1, if_neg h2]\n rfl\n done\n\nlemma swap_values (u v i : Nat) : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := by\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n apply Or.inl\n rewrite [h1]\n show swap u v u = v from swap_fst u v\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n apply Or.inr\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n apply Or.inl\n rewrite [h2]\n show swap u v v = u from swap_snd u v\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n apply Or.inr\n show swap u v i = i from swap_other h1 h2\n done\n done\n done\n\nlemma swap_maps_below {u v n : Nat} (h1 : u < n) (h2 : v < n) : maps_below n (swap u v) := by\n define\n fix i : Nat\n assume h3 : i < n\n have h4 : swap u v i = v \u2228 swap u v i = u \u2228 swap u v i = i := swap_values u v i\n by_cases on h4\n \u00b7 -- Case 1. h4 : swap u v i = v\n rewrite [h4]\n show v < n from h2\n done\n \u00b7 -- Case 2.\n by_cases on h4\n \u00b7 -- Case 2.1. h4 : swap u v i = u\n rewrite [h4]\n show u < n from h1\n done\n \u00b7 -- Case 2.2. h4 : swap u v i = i\n rewrite [h4]\n show i < n from h3\n done\n done\n done\n\nlemma swap_swap (u v n : Nat) : \u2200 i < n, swap u v (swap u v i) = i := by\n fix i : Nat\n assume h : i < n\n by_cases h1 : i = u\n \u00b7 -- Case 1. h1 : i = u\n rewrite [h1, swap_fst, swap_snd]\n rfl\n done\n \u00b7 -- Case 2. h1 : i \u2260 u\n by_cases h2 : i = v\n \u00b7 -- Case 2.1. h2 : i = v\n rewrite [h2, swap_snd, swap_fst]\n rfl\n done\n \u00b7 -- Case 2.2. h2 : i \u2260 v\n rewrite [swap_other h1 h2, swap_other h1 h2]\n rfl\n done\n done\n done\n\nlemma swap_one_one_below (u v n) : one_one_below n (swap u v) :=\n left_inv_one_one_below (swap_swap u v n)\n\nlemma swap_onto_below {u v n} (h1 : u < n) (h2 : v < n) : onto_below n (swap u v) :=\n right_inv_onto_below (swap_swap u v n) (swap_maps_below h1 h2)\n\nlemma swap_perm_below {u v n} (h1 : u < n) (h2 : v < n) : perm_below n (swap u v) :=\n And.intro (swap_maps_below h1 h2) (And.intro (swap_one_one_below u v n) (swap_onto_below h1 h2))\n\nlemma comp_perm_below {n : Nat} {f g : Nat \u2192 Nat}\n (h1 : perm_below n f) (h2 : perm_below n g) :\n perm_below n (f \u2218 g) := sorry\n\nlemma trivial_swap (u : Nat) : swap u u = id := by\n apply funext\n fix x : Nat\n by_cases h1 : x = u\n \u00b7 -- Case 1. h1 : x = u\n rewrite [h1, swap_fst]\n rfl\n done\n \u00b7 -- Case 2. h1 : x \u2260 u\n rewrite [swap_other h1 h1]\n rfl\n done\n done\n\nlemma prod_eq_fun {m : Nat} (f g : Nat \u2192 ZMod m) (k : Nat) :\n \u2200 (n : Nat), (\u2200 i < n, f (k + i) = g (k + i)) \u2192\n prod_seq n k f = prod_seq n k g := by\n by_induc\n \u00b7 -- Base Case\n assume h : (\u2200 i < 0, f (k + i) = g (k + i))\n rewrite [prod_seq_base, prod_seq_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (\u2200 i < n, f (k + i) = g (k + i)) \u2192 prod_seq n k f = prod_seq n k g\n assume h1 : \u2200 i < n + 1, f (k + i) = g (k + i)\n have h2 : \u2200 i < n, f (k + i) = g (k + i) := by\n fix i : Nat\n assume h2 : i < n\n have h3 : i < n + 1 := by linarith\n show f (k + i) = g (k + i) from h1 i h3\n done\n have h3 : prod_seq n k f = prod_seq n k g := ih h2\n have h4 : n < n + 1 := Nat.lt_succ_self n\n rewrite [prod_seq_step, prod_seq_step, h3, h1 n h4]\n rfl\n done\n done\n\nlemma swap_prod_eq_prod_below {m u n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : u \u2264 n) : prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f := by\n have h2 : \u2200 (i : Nat), i < u \u2192 (f \u2218 swap u n) (0 + i) = f (0 + i) := by\n fix i : Nat\n assume h2 : i < u\n have h3 : 0 + i \u2260 u := by linarith\n have h4 : 0 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n done\n show prod_seq u 0 (f \u2218 swap u n) = prod_seq u 0 f from\n prod_eq_fun (f \u2218 swap u n) f 0 u h2\n done\n\nlemma swap_prod_eq_prod_between {m u j n : Nat} (f : Nat \u2192 ZMod m)\n (h1 : n = u + 1 + j) : prod_seq j (u + 1) (f \u2218 swap u n) =\n prod_seq j (u + 1) f := by\n have h2 : \u2200 i < j, (f \u2218 swap u n) (u + 1 + i) = f (u + 1 + i) := by\n fix i : Nat\n assume h2 : i < j\n have h3 : u + 1 + i \u2260 u := by linarith\n have h4 : u + 1 + i \u2260 n := by linarith\n rewrite [comp_def, swap_other h3 h4]\n rfl\n show prod_seq j (u + 1) (f \u2218 swap u n) = prod_seq j (u + 1) f from\n prod_eq_fun (f \u2218 swap u n) f (u + 1) j h2\n done\n\n", "theoremStatement": "lemma break_prod {m : Nat} (n : Nat) (f : Nat \u2192 ZMod m) :\n \u2200 (j : Nat), prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f ", "theoremName": "HTPI.break_prod", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": 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"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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n by_induc\n \u00b7 -- Base Case\n have h : n + 0 = n := by rfl\n rewrite [prod_seq_base, h, Theorem_7_3_6_7]\n rfl\n done\n \u00b7 -- Induction Step\n fix j : Nat\n assume ih : prod_seq (n + j) 0 f = prod_seq n 0 f * prod_seq j n f\n rewrite [\u2190add_assoc, prod_seq_zero_step, prod_seq_step, ih, mul_assoc]\n rfl\n done\n done", "proofType": "tactic", "proofLengthLines": 13, "proofLengthTokens": 339}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.IntroLean\nnamespace HTPI\n\n/- Definitions -/\ndef even (n : Int) : Prop := \u2203 (k : Int), n = 2 * k\n\ndef odd (n : Int) : Prop := \u2203 (k : Int), n = 2 * k + 1\n\n/- Sections 3.1 and 3.2 -/\ntheorem Example_3_2_4_v2 (P Q R : Prop)\n (h : P \u2192 (Q \u2192 R)) : \u00acR \u2192 (P \u2192 \u00acQ) := by\n assume h2 : \u00acR\n assume h3 : P\n by_contra h4\n have h5 : Q \u2192 R := h h3\n have h6 : R := h5 h4\n show False from h2 h6\n done\n\ntheorem Example_3_2_4_v3 (P Q R : Prop)\n (h : P \u2192 (Q \u2192 R)) : \u00acR \u2192 (P \u2192 \u00acQ) := by\n assume h2 : \u00acR\n assume h3 : P\n by_contra h4\n contradict h2\n show R from h h3 h4\n done\n\ntheorem Like_Example_3_2_5\n (U : Type) (A B C : Set U) (a : U)\n (h1 : a \u2208 A) (h2 : a \u2209 A \\ B)\n (h3 : a \u2208 B \u2192 a \u2208 C) : a \u2208 C := by\n apply h3 _\n define at h2\n demorgan at h2; conditional at h2\n show a \u2208 B from h2 h1\n done\n\n/- Section 3.3 -/\nexample (U : Type) (P Q : Pred U)\n (h1 : \u2200 (x : U), P x \u2192 \u00acQ x)\n (h2 : \u2200 (x : U), Q x) :\n \u00ac\u2203 (x : U), P x := by\n quant_neg --Goal is now \u2200 (x : U), \u00acP x\n fix y : U\n have h3 : P y \u2192 \u00acQ y := h1 y\n have h4 : Q y := h2 y\n contrapos at h3 --Now h3 : Q y \u2192 \u00acP y\n show \u00acP y from h3 h4\n done\n\nexample (U : Type) (A B C : Set U) (h1 : A \u2286 B \u222a C)\n (h2 : \u2200 (x : U), x \u2208 A \u2192 x \u2209 B) : A \u2286 C := by\n define --Goal : \u2200 \u2983a : U\u2984, a \u2208 A \u2192 a \u2208 C\n fix y : U\n assume h3 : y \u2208 A\n have h4 : y \u2209 B := h2 y h3\n define at h1 --h1 : \u2200 \u2983a : U\u2984, a \u2208 A \u2192 a \u2208 B \u222a C\n have h5 : y \u2208 B \u222a C := h1 h3\n define at h5 --h5 : y \u2208 B \u2228 y \u2208 C\n conditional at h5 --h5 : \u00acy \u2208 B \u2192 y \u2208 C\n show y \u2208 C from h5 h4\n done\n\nexample (U : Type) (P Q : Pred U)\n (h1 : \u2200 (x : U), \u2203 (y : U), P x \u2192 \u00ac Q y)\n (h2 : \u2203 (x : U), \u2200 (y : U), P x \u2192 Q y) :\n \u2203 (x : U), \u00acP x := by\n obtain (a : U)\n (h3 : \u2200 (y : U), P a \u2192 Q y) from h2\n have h4 : \u2203 (y : U), P a \u2192 \u00ac Q y := h1 a\n obtain (b : U) (h5 : P a \u2192 \u00ac Q b) from h4\n have h6 : P a \u2192 Q b := h3 b\n apply Exists.intro a _\n by_contra h7\n show False from h5 h7 (h6 h7)\n done\n\ntheorem Example_3_3_5 (U : Type) (B : Set U)\n (F : Set (Set U)) : \u22c3\u2080 F \u2286 B \u2192 F \u2286 \ud835\udcab B := by\n assume h1 : \u22c3\u2080 F \u2286 B\n define\n fix x : Set U\n assume h2 : x \u2208 F\n define\n fix y : U\n assume h3 : y \u2208 x\n define at h1\n apply h1 _\n define\n apply Exists.intro x _\n show x \u2208 F \u2227 y \u2208 x from And.intro h2 h3\n done\n\n/- Section 3.4 -/\ntheorem Like_Example_3_4_1 (U : Type)\n (A B C D : Set U) (h1 : A \u2286 B)\n (h2 : \u00ac\u2203 (c : U), c \u2208 C \u2229 D) :\n A \u2229 C \u2286 B \\ D := by\n define\n fix x : U\n assume h3 : x \u2208 A \u2229 C\n define at h3; define\n apply And.intro\n \u00b7 -- Proof that x \u2208 B.\n show x \u2208 B from h1 h3.left\n done\n \u00b7 -- Proof that x \u2209 D.\n contradict h2 with h4\n apply Exists.intro x\n show x \u2208 C \u2229 D from And.intro h3.right h4\n done\n done\n\nexample (U : Type) (P Q : Pred U)\n (h1 : \u2200 (x : U), P x \u2194 Q x) :\n (\u2203 (x : U), P x) \u2194 \u2203 (x : U), Q x := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : \u2203 (x : U), P x\n obtain (u : U) (h3 : P u) from h2\n have h4 : P u \u2194 Q u := h1 u\n apply Exists.intro u\n show Q u from h4.ltr h3\n done\n \u00b7 -- (\u2190)\n assume h2 : \u2203 (x : U), Q x\n obtain (u : U) (h3 : Q u) from h2\n show \u2203 (x : U), P x from Exists.intro u ((h1 u).rtl h3)\n done\n done\n\ntheorem Example_3_4_5 (U : Type)\n (A B C : Set U) : A \u2229 (B \\ C) = (A \u2229 B) \\ C := by\n apply Set.ext\n fix x : U\n show x \u2208 A \u2229 (B \\ C) \u2194 x \u2208 (A \u2229 B) \\ C from\n calc x \u2208 A \u2229 (B \\ C)\n _ \u2194 x \u2208 A \u2227 (x \u2208 B \u2227 x \u2209 C) := Iff.refl _\n _ \u2194 (x \u2208 A \u2227 x \u2208 B) \u2227 x \u2209 C := and_assoc.symm\n _ \u2194 x \u2208 (A \u2229 B) \\ C := Iff.refl _\n done\n\n/- Section 3.5 -/\ntheorem Example_3_5_2\n (U : Type) (A B C : Set U) :\n A \\ (B \\ C) \u2286 (A \\ B) \u222a C := by\n fix x : U\n assume h1 : x \u2208 A \\ (B \\ C)\n define; define at h1\n have h2 : x \u2209 B \\ C := h1.right\n define at h2; demorgan at h2\n --h2 : x \u2209 B \u2228 x \u2208 C\n by_cases on h2\n \u00b7 -- Case 1. h2 : x \u2209 B\n apply Or.inl\n show x \u2208 A \\ B from And.intro h1.left h2\n done\n \u00b7 -- Case 2. h2 : x \u2208 C\n apply Or.inr\n show x \u2208 C from h2\n done\n done\n\nexample (U : Type) (A B C : Set U)\n (h1 : A \\ B \u2286 C) : A \u2286 B \u222a C := by\n fix x : U\n assume h2 : x \u2208 A\n define\n or_right with h3\n show x \u2208 C from h1 (And.intro h2 h3)\n done\n\nexample\n (U : Type) (A B C : Set U) (h1 : A \u2286 B \u222a C)\n (h2 : \u00ac\u2203 (x : U), x \u2208 A \u2229 B) : A \u2286 C := by\n fix a : U\n assume h3 : a \u2208 A\n quant_neg at h2\n have h4 : a \u2208 B \u222a C := h1 h3\n have h5 : a \u2209 A \u2229 B := h2 a\n define at h4\n define at h5; demorgan at h5\n disj_syll h5 h3 --h5 : \u00aca \u2208 B\n disj_syll h4 h5 --h4 : a \u2208 C\n show a \u2208 C from h4\n done\n\nexample\n (U : Type) (A B C : Set U) (h1 : A \u2286 B \u222a C)\n (h2 : \u00ac\u2203 (x : U), x \u2208 A \u2229 B) : A \u2286 C := by\n fix a : U\n assume h3 : a \u2208 A\n have h4 : a \u2208 B \u222a C := h1 h3\n define at h4\n have h5 : a \u2209 B := by\n contradict h2 with h6\n show \u2203 (x : U), x \u2208 A \u2229 B from\n Exists.intro a (And.intro h3 h6)\n done\n disj_syll h4 h5 --h4 : a \u2208 C\n show a \u2208 C from h4\n done\n\n/- Section 3.6 -/\ntheorem empty_union {U : Type} (B : Set U) :\n \u2205 \u222a B = B := by\n apply Set.ext\n fix x : U\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : x \u2208 \u2205 \u222a B\n define at h1\n have h2 : x \u2209 \u2205 := by\n by_contra h3\n define at h3 --h3 : False\n show False from h3\n done\n disj_syll h1 h2 --h1 : x \u2208 B\n show x \u2208 B from h1\n done\n \u00b7 -- (\u2190)\n assume h1 : x \u2208 B\n show x \u2208 \u2205 \u222a B from Or.inr h1\n done\n done\n\ntheorem union_comm {U : Type} (X Y : Set U) :\n X \u222a Y = Y \u222a X := by\n apply Set.ext\n fix x : U\n define : x \u2208 X \u222a Y\n define : x \u2208 Y \u222a X\n show x \u2208 X \u2228 x \u2208 Y \u2194 x \u2208 Y \u2228 x \u2208 X from or_comm\n done\n\ntheorem Example_3_6_2 (U : Type) :\n \u2203! (A : Set U), \u2200 (B : Set U),\n A \u222a B = B := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro \u2205\n show \u2200 (B : Set U), \u2205 \u222a B = B from empty_union\n done\n \u00b7 -- Uniqueness\n fix C : Set U; fix D : Set U\n assume h1 : \u2200 (B : Set U), C \u222a B = B\n assume h2 : \u2200 (B : Set U), D \u222a B = B\n have h3 : C \u222a D = D := h1 D\n have h4 : D \u222a C = C := h2 C\n show C = D from\n calc C\n _ = D \u222a C := h4.symm\n _ = C \u222a D := union_comm D C\n _ = D := h3\n done\n done\n\ntheorem Example_3_6_4 (U : Type) (A B C : Set U)\n (h1 : \u2203 (x : U), x \u2208 A \u2229 B)\n (h2 : \u2203 (x : U), x \u2208 A \u2229 C)\n (h3 : \u2203! (x : U), x \u2208 A) :\n \u2203 (x : U), x \u2208 B \u2229 C := by\n obtain (b : U) (h4 : b \u2208 A \u2229 B) from h1\n obtain (c : U) (h5 : c \u2208 A \u2229 C) from h2\n obtain (a : U) (h6 : a \u2208 A) (h7 : \u2200 (y z : U),\n y \u2208 A \u2192 z \u2208 A \u2192 y = z) from h3\n define at h4; define at h5\n have h8 : b = c := h7 b c h4.left h5.left\n rewrite [h8] at h4\n show \u2203 (x : U), x \u2208 B \u2229 C from\n Exists.intro c (And.intro h4.right h5.right)\n done\n\n/- Section 3.7 -/\ntheorem Theorem_3_3_7 :\n \u2200 (a b c : Int), a \u2223 b \u2192 b \u2223 c \u2192 a \u2223 c := by\n fix a : Int; fix b : Int; fix c : Int\n assume h1 : a \u2223 b; assume h2 : b \u2223 c\n define at h1; define at h2; define\n obtain (m : Int) (h3 : b = a * m) from h1\n obtain (n : Int) (h4 : c = b * n) from h2\n rewrite [h3] at h4 --h4 : c = a * m * n\n apply Exists.intro (m * n)\n rewrite [mul_assoc a m n] at h4\n show c = a * (m * n) from h4\n done\n\ntheorem Theorem_3_4_7 :\n \u2200 (n : Int), 6 \u2223 n \u2194 2 \u2223 n \u2227 3 \u2223 n := by\n fix n : Int\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : 6 \u2223 n; define at h1\n obtain (k : Int) (h2 : n = 6 * k) from h1\n apply And.intro\n \u00b7 -- Proof that 2 \u2223 n\n define\n apply Exists.intro (3 * k)\n rewrite [\u2190mul_assoc 2 3 k]\n show n = 2 * 3 * k from h2\n done\n \u00b7 -- Proof that 3 \u2223 n\n define\n apply Exists.intro (2 * k)\n rewrite [\u2190mul_assoc 3 2 k]\n show n = 3 * 2 * k from h2\n done\n done\n \u00b7 -- (\u2190)\n assume h1 : 2 \u2223 n \u2227 3 \u2223 n\n have h2 : 2 \u2223 n := h1.left\n have h3 : 3 \u2223 n := h1.right\n define at h2; define at h3; define\n obtain (j : Int) (h4 : n = 2 * j) from h2\n obtain (k : Int) (h5 : n = 3 * k) from h3\n have h6 : 6 * (j - k) = n :=\n calc 6 * (j - k)\n _ = 3 * (2 * j) - 2 * (3 * k) := by ring\n _ = 3 * n - 2 * n := by rw [\u2190h4, \u2190h5]\n _ = n := by ring\n show \u2203 (c : Int), n = 6 * c from\n Exists.intro (j - k) h6.symm\n done\n done\n\n", "theoremStatement": "theorem Example_3_5_4 (x : Real) (h1 : x \u2264 x ^ 2) : x \u2264 0 \u2228 1 \u2264 x ", "theoremName": "HTPI.Example_3_5_4", "fileCreated": {"commit": "cc48e6ba4c656bf74f7690323b2c27f2c466665b", "date": "2023-03-11"}, "theoremCreated": {"commit": "515de92e6c12bba8178aad612650f80eacd0a245", "date": "2023-04-13"}, "file": "htpi/HTPILib/Chap3.lean", "module": "HTPILib.Chap3", "jsonFile": "HTPILib.Chap3.jsonl", "positionMetadata": {"lineInFile": 329, "tokenPositionInFile": 8096, "theoremPositionInFile": 15}, "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", 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"Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n or_right with h2 --h2 : \u00acx \u2264 0; Goal : 1 \u2264 x\n have h3 : 0 < x := lt_of_not_le h2\n have h4 : 1 * x \u2264 x * x :=\n calc 1 * x\n _ = x := one_mul x\n _ \u2264 x ^ 2 := h1\n _ = x * x := by ring\n show 1 \u2264 x from le_of_mul_le_mul_right h4 h3\n done", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 266}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap7\nnamespace HTPI\n\n/- Definitions -/\ndef fnz (n : Nat) : Int := if 2 \u2223 n then \u2191(n / 2) else -\u2191((n + 1) / 2)\n\ndef fzn (a : Int) : Nat := if a \u2265 0 then 2 * Int.toNat a else 2 * Int.toNat (-a) - 1\n\ndef tri (k : Nat) : Nat := k * (k + 1) / 2\n\ndef fnnn (p : Nat \u00d7 Nat) : Nat := tri (p.1 + p.2) + p.1\n\ndef num_elts_below (A : Set Nat) (m s : Nat) : Prop :=\n match m with\n | 0 => s = 0\n | n + 1 => (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s)\n\ndef enum (A : Set Nat) (s n : Nat) : Prop := n \u2208 A \u2227 num_elts_below A n s\n\ndef unique_val_on_N {U : Type} (R : Rel Nat U) : Prop :=\n \u2200 \u2983n : Nat\u2984 \u2983x1 x2 : U\u2984, R n x1 \u2192 R n x2 \u2192 x1 = x2\n\ndef nat_rel_onto {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203 (n : Nat), R n x\n\ndef fcnl_onto_from_nat {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n unique_val_on_N R \u2227 nat_rel_onto R A\n\ndef fcnl_one_one_to_nat {U : Type} (R : Rel U Nat) (A : Set U) : Prop :=\n fcnl_on R A \u2227 \u2200 \u2983x1 x2 : U\u2984 \u2983n : Nat\u2984,\n (x1 \u2208 A \u2227 R x1 n) \u2192 (x2 \u2208 A \u2227 R x2 n) \u2192 x1 = x2\n\ndef least_rel_to {U : Type} (S : Rel Nat U) (x : U) (n : Nat) : Prop :=\n S n x \u2227 \u2200 (m : Nat), S m x \u2192 n \u2264 m\n\ndef restrict_to {U V : Type} (S : Rel U V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 S x y\n\ndef fqn (q : Rat) : Nat := fnnn (fzn q.num, q.den)\n\ndef Set_rp_below (m : Nat) : Set Nat := {n : Nat | rel_prime m n \u2227 n < m}\n\ndef Set_prod {U V : Type} (A : Set U) (B : Set V) : Set (U \u00d7 V) :=\n {(a, b) : U \u00d7 V | a \u2208 A \u2227 b \u2208 B}\n\nnotation:75 A:75 \" \u00d7\u209b \" B:75 => Set_prod A B\n\ndef Rel_prod {U V W X : Type} (R : Rel U V) (S : Rel W X)\n (p : U \u00d7 W) (q : V \u00d7 X) : Prop := R p.1 q.1 \u2227 S p.2 q.2\n\nnotation:75 R:75 \" \u00d7\u1d63 \" S:75 => Rel_prod R S\n\ndef qr (n a : Nat) : Nat \u00d7 Nat := (a / n, a % n)\n\ndef mod_mod (m n a : Nat) : Nat \u00d7 Nat := (a % m, a % n)\n\ndef enum_union_fam {U : Type}\n (F : Set (Set U)) (f : Set U \u2192 Rel Nat U) (R : Rel Nat (Set U))\n (n : Nat) (a : U) : Prop := \u2203 (p : Nat \u00d7 Nat), fnnn p = n \u2227\n \u2203 A \u2208 F, R p.1 A \u2227 (f A) p.2 a\n\ndef seq {U : Type} (A : Set U) : Set (List U) :=\n {l : List U | \u2200 x \u2208 l, x \u2208 A}\n\ndef seq_by_length {U : Type} (A : Set U) (n : Nat) : Set (List U) :=\n {l : List U | l \u2208 seq A \u2227 l.length = n}\n\ndef seq_cons (U : Type) (p : U \u00d7 (List U)) : List U := p.1 :: p.2\n\ndef sbl_set {U : Type} (A : Set U) : Set (Set (List U)) :=\n {S : Set (List U) | \u2203 (n : Nat), seq_by_length A n = S}\n\ndef rep_common_image\n {U V : Type} (R S : Rel U V) (X0 : Set U) (n : Nat) : Set U :=\n match n with\n | 0 => X0\n | m + 1 => {a : U | \u2203 x \u2208 rep_common_image R S X0 m,\n \u2203 (y : V), R x y \u2227 S a y}\n\ndef cum_rep_image {U V : Type} (R S : Rel U V) (X0 : Set U) : Set U :=\n {a : U | \u2203 (n : Nat), a \u2208 rep_common_image R S X0 n}\n\ndef csb_match {U V : Type} (R S : Rel U V) (X0 : Set U)\n (x : U) (y : V) : Prop := x \u2208 cum_rep_image R S X0 \u2227 R x y \u2228\n x \u2209 cum_rep_image R S X0 \u2227 S x y\n\n/- Section 8.1 -/\n#eval [fnz 0, fnz 1, fnz 2, fnz 3, fnz 4, fnz 5, fnz 6]\n --Answer: [0, -1, 1, -2, 2, -3, 3]\n\n#eval [fzn 0, fzn (-1), fzn 1, fzn (-2), fzn 2, fzn (-3), fzn 3]\n --Answer: [0, 1, 2, 3, 4, 5, 6]\n\nlemma fnz_even (k : Nat) : fnz (2 * k) = \u2191k := by\n have h1 : 2 \u2223 2 * k := by\n apply Exists.intro k\n rfl\n done\n have h2 : fnz (2 * k) = if 2 \u2223 2 * k then \u2191(2 * k / 2)\n else -\u2191((2 * k + 1) / 2) := by rfl\n rewrite [if_pos h1] at h2 --h2 : fnz (2 * k) = \u2191(2 * k / 2)\n have h3 : 0 < 2 := by linarith\n rewrite [Nat.mul_div_cancel_left k h3] at h2\n show fnz (2 * k) = \u2191k from h2\n done\n\nlemma fnz_odd (k : Nat) : fnz (2 * k + 1) = -\u2191(k + 1) := sorry\n\nlemma fzn_nat (k : Nat) : fzn \u2191k = 2 * k := by rfl\n\nlemma fzn_neg_succ_nat (k : Nat) : fzn (-\u2191(k + 1)) = 2 * k + 1 := by rfl\n\n--From exercises of Section 6.1\ntheorem Exercise_6_1_16a1 : \u2200 (n : Nat), nat_even n \u2228 nat_odd n := sorry\n\nlemma fzn_fnz : fzn \u2218 fnz = id := by\n apply funext --Goal : \u2200 (x : Nat), (fzn \u2218 fnz) x = id x\n fix n : Nat\n rewrite [comp_def] --Goal : fzn (fnz n) = id n\n have h1 : nat_even n \u2228 nat_odd n := Exercise_6_1_16a1 n\n by_cases on h1\n \u00b7 -- Case 1. h1 : nat_even n\n obtain (k : Nat) (h2 : n = 2 * k) from h1\n rewrite [h2, fnz_even, fzn_nat]\n rfl\n done\n \u00b7 -- Case 2. h1 : nat_odd n\n obtain (k : Nat) (h2 : n = 2 * k + 1) from h1\n rewrite [h2, fnz_odd, fzn_neg_succ_nat]\n rfl\n done\n done\n\nlemma fnz_fzn : fnz \u2218 fzn = id := sorry\n\nlemma fzn_one_one : one_to_one fzn := Theorem_5_3_3_1 fzn fnz fnz_fzn\n\nlemma fzn_onto : onto fzn := Theorem_5_3_3_2 fzn fnz fzn_fnz\n\nlemma fnz_one_one : one_to_one fnz := Theorem_5_3_3_1 fnz fzn fzn_fnz\n\nlemma fnz_onto : onto fnz := Theorem_5_3_3_2 fnz fzn fnz_fzn\n\nlemma fnnn_def (a b : Nat) : fnnn (a, b) = tri (a + b) + a := by rfl\n\n#eval [fnnn (0, 0), fnnn (0, 1), fnnn (1, 0), fnnn (0, 2), fnnn (1, 1)]\n --Answer: [0, 1, 2, 3, 4]\n\nlemma tri_step (k : Nat) : tri (k + 1) = tri k + k + 1 := sorry\n\nlemma tri_incr {j k : Nat} (h1 : j \u2264 k) : tri j \u2264 tri k := sorry\n\nlemma le_of_fnnn_eq {a1 b1 a2 b2 : Nat}\n (h1 : fnnn (a1, b1) = fnnn (a2, b2)) : a1 + b1 \u2264 a2 + b2 := by\n by_contra h2\n have h3 : a2 + b2 + 1 \u2264 a1 + b1 := by linarith\n have h4 : fnnn (a2, b2) < fnnn (a1, b1) :=\n calc fnnn (a2, b2)\n _ = tri (a2 + b2) + a2 := by rfl\n _ < tri (a2 + b2) + (a2 + b2) + 1 := by linarith\n _ = tri (a2 + b2 + 1) := (tri_step _).symm\n _ \u2264 tri (a1 + b1) := tri_incr h3\n _ \u2264 tri (a1 + b1) + a1 := by linarith\n _ = fnnn (a1, b1) := by rfl\n linarith\n done\n\nlemma fnnn_one_one : one_to_one fnnn := by\n fix (a1, b1) : Nat \u00d7 Nat\n fix (a2, b2) : Nat \u00d7 Nat\n assume h1 : fnnn (a1, b1) = fnnn (a2, b2) --Goal : (a1, b1) = (a2, b2)\n have h2 : a1 + b1 \u2264 a2 + b2 := le_of_fnnn_eq h1\n have h3 : a2 + b2 \u2264 a1 + b1 := le_of_fnnn_eq h1.symm\n have h4 : a1 + b1 = a2 + b2 := by linarith\n rewrite [fnnn_def, fnnn_def, h4] at h1\n --h1 : tri (a2 + b2) + a1 = tri (a2 + b2) + a2\n have h6 : a1 = a2 := Nat.add_left_cancel h1\n rewrite [h6] at h4 --h4 : a2 + b1 = a2 + b2\n have h7 : b1 = b2 := Nat.add_left_cancel h4\n rewrite [h6, h7]\n rfl\n done\n\nlemma fnnn_onto : onto fnnn := by\n define --Goal : \u2200 (y : Nat), \u2203 (x : Nat \u00d7 Nat), fnnn x = y\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro (0, 0)\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2203 (x : Nat \u00d7 Nat), fnnn x = n\n obtain ((a, b) : Nat \u00d7 Nat) (h1 : fnnn (a, b) = n) from ih\n by_cases h2 : b = 0\n \u00b7 -- Case 1. h2 : b = 0\n apply Exists.intro (0, a + 1)\n show fnnn (0, a + 1) = n + 1 from\n calc fnnn (0, a + 1)\n _ = tri (0 + (a + 1)) + 0 := by rfl\n _ = tri (a + 1) := by ring\n _ = tri a + a + 1 := tri_step a\n _ = tri (a + 0) + a + 1 := by ring\n _ = fnnn (a, b) + 1 := by rw [h2, fnnn_def]\n _ = n + 1 := by rw [h1]\n done\n \u00b7 -- Case 2. h2 : b \u2260 0\n obtain (k : Nat) (h3 : b = k + 1) from\n exists_eq_add_one_of_ne_zero h2\n apply Exists.intro (a + 1, k)\n show fnnn (a + 1, k) = n + 1 from\n calc fnnn (a + 1, k)\n _ = tri (a + 1 + k) + (a + 1) := by rfl\n _ = tri (a + (k + 1)) + a + 1 := by ring\n _ = tri (a + b) + a + 1 := by rw [h3]\n _ = fnnn (a, b) + 1 := by rfl\n _ = n + 1 := by rw [h1]\n done\n done\n done\n\n", "theoremStatement": "lemma one_one_on_of_one_one {U V : Type} {f : U \u2192 V}\n (h : one_to_one f) (A : Set U) : one_one_on f A ", "theoremName": "HTPI.one_one_on_of_one_one", "fileCreated": {"commit": "fe2721d65e0771872fe4cb8117517e643ac1dc1f", "date": "2023-05-10"}, "theoremCreated": {"commit": "01538f73db4cc69cc47bd51e795bf25d251c4b6d", "date": "2023-05-31"}, "file": "htpi/HTPILib/Chap8Part2.lean", "module": "HTPILib.Chap8Part2", "jsonFile": "HTPILib.Chap8Part2.jsonl", "positionMetadata": {"lineInFile": 225, "tokenPositionInFile": 7314, "theoremPositionInFile": 45}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 13, "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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"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", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "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", 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"Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "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.Meta.Tactic.Delta", "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.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "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.Meta.Tactic.Symm", "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.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "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", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6", "HTPILib.Chap7"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n define\n fix x1 : U; fix x2 : U\n assume h1 : x1 \u2208 A\n assume h2 : x2 \u2208 A\n show f x1 = f x2 \u2192 x1 = x2 from h x1 x2\n done", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 130}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\n", "theoremStatement": "lemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m ", "theoremName": "HTPI.cc_eq_iff_sub_zero", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 850, "tokenPositionInFile": 26322, "theoremPositionInFile": 83}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 88, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", 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"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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done", "proofType": "tactic", "proofLengthLines": 26, "proofLengthTokens": 800}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\n", "theoremStatement": "lemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i ", "theoremName": "HTPI.Ginv_right_inv", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1324, "tokenPositionInFile": 42902, "theoremPositionInFile": 127}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 6, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 33, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", 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"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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 327}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := 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"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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 200}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap8Part1\nnamespace HTPI\n\n/- Definitions and theorems in Chap8Part1 and HTPIDefs\n\ntheorem zero_elts_iff_empty {A : Type} (X : Set A) :\n numElts X 0 \u2194 empty X\n\ntheorem one_elt_iff_singleton {A : Type} (X : Set A) :\n numElts X 1 \u2194 \u2203 (x : A), X = {x}\n\ntheorem nonempty_of_pos_numElts {A : Type} {X : Set A} {n : Nat}\n (h1 : numElts X n) (h2 : n > 0) : \u2203 (x : A), x \u2208 X\n\ntheorem remove_one_numElts {A : Type} {X : Set A} {n : Nat} {x : A}\n (h1 : numElts X (n + 1)) (h2 : x \u2208 X) : numElts (X \\ {x}) n\n\ndef sum_seq {A : Type} [AddZeroClass A] (m k : Nat) (f : Nat \u2192 A) : A :=\n match m with\n | 0 => 0\n | n + 1 => sum_seq n k f + f (k + n)\n\ntheorem sum_base {A : Type} [AddZeroClass A] {k : Nat} {f : Nat \u2192 A} :\n Sum i from k to k, f i = f k\n\ntheorem sum_step {A : Type} [AddZeroClass A] {k n : Nat} {f : Nat \u2192 A}\n (h : k \u2264 n) : Sum i from k to (n + 1), f i = (Sum i from k to n, f i) + f (n + 1)\n\ntheorem sum_from_zero_step {A : Type} [AddZeroClass A] {n : Nat} {f : Nat \u2192 A} :\n Sum i from 0 to (n + 1), f i = (Sum i from 0 to n, f i) + f (n + 1)\n-/\n\n/- Definitions -/\ndef nat_even (n : Nat) : Prop := \u2203 (k : Nat), n = 2 * k\n\ndef nat_odd (n : Nat) : Prop := \u2203 (k : Nat), n = 2 * k + 1\n\ndef extendPO {A : Type} (R : BinRel A) (b : A) (x y : A) : Prop :=\n R x y \u2228 (R x b \u2227 \u00acR y b)\n\ndef fact (k : Nat) : Nat :=\n match k with\n | 0 => 1\n | n + 1 => (n + 1) * fact n\n\ndef Fib (n : Nat) : Nat :=\n match n with\n | 0 => 0\n | 1 => 1\n | k + 2 => Fib k + Fib (k + 1)\n\ndef rep_image {A : Type} (f : A \u2192 A) (n : Nat) (B : Set A) : Set A :=\n match n with\n | 0 => B\n | k + 1 => image f (rep_image f k B)\n\ndef cumul_image {A : Type} (f : A \u2192 A) (B : Set A) : Set A :=\n {x : A | \u2203 (n : Nat), x \u2208 rep_image f n B}\n\n/- Section 6.1 -/\ntheorem Like_Example_6_1_2 :\n \u2200 (n : Nat), 3 \u2223 n ^ 3 + 2 * n := by\n by_induc\n \u00b7 -- Base Case\n define --Goal : \u2203 (c : Nat), 0 ^ 3 + 2 * 0 = 3 * c\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : 3 \u2223 n ^ 3 + 2 * n\n define at ih --ih : \u2203 (c : Nat), n ^ 3 + 2 * n = 3 * c\n obtain (k : Nat) (h1 : n ^ 3 + 2 * n = 3 * k) from ih\n define --Goal : \u2203 (c : Nat), (n + 1) ^ 3 + 2 * (n + 1) = 3 * c\n apply Exists.intro (k + n ^ 2 + n + 1)\n show (n + 1) ^ 3 + 2 * (n + 1) = 3 * (k + n ^ 2 + n + 1) from\n calc (n + 1) ^ 3 + 2 * (n + 1)\n _ = n ^ 3 + 2 * n + 3 * n ^ 2 + 3 * n + 3 := by ring\n _ = 3 * k + 3 * n ^ 2 + 3 * n + 3 := by rw [h1]\n _ = 3 * (k + n ^ 2 + n + 1) := by ring\n done\n done\n\ntheorem Like_Example_6_1_1 :\n \u2200 (n : Nat), (Sum i from 0 to n, 2 ^ i) + 1 = 2 ^ (n + 1) := by\n by_induc\n \u00b7 -- Base Case\n rewrite [sum_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (Sum i from 0 to n, 2 ^ i) + 1 = 2 ^ (n + 1)\n show (Sum i from 0 to n + 1, 2 ^ i) + 1 = 2 ^ (n + 1 + 1) from\n calc (Sum i from 0 to n + 1, 2 ^ i) + 1\n _ = (Sum i from 0 to n, 2 ^ i) + 2 ^ (n + 1) + 1 := by\n rw [sum_from_zero_step]\n _ = (Sum i from 0 to n, 2 ^ i) + 1 + 2 ^ (n + 1) := by ring\n _ = 2 ^ (n + 1) + 2 ^ (n + 1) := by rw [ih]\n _ = 2 ^ (n + 1 + 1) := by ring\n done\n done\n\ntheorem Example_6_1_3 : \u2200 n \u2265 5, 2 ^ n > n ^ 2 := by\n by_induc\n \u00b7 -- Base Case\n decide\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume h1 : n \u2265 5\n assume ih : 2 ^ n > n ^ 2\n have h2 : n * n \u2265 5 * n := Nat.mul_le_mul_right n h1\n show 2 ^ (n + 1) > (n + 1) ^ 2 from\n calc 2 ^ (n + 1)\n _ = 2 * 2 ^ n := by ring\n _ > 2 * n ^ 2 := by linarith\n _ \u2265 n ^ 2 + 5 * n := by linarith\n _ > n ^ 2 + 2 * n + 1 := by linarith\n _ = (n + 1) ^ 2 := by ring\n done\n done\n\n#eval 2 - 3 --Answer: 0\n\ntheorem Example_6_1_1 :\n \u2200 (n : Nat), Sum i from 0 to n, (2 : Int) ^ i =\n (2 : Int) ^ (n + 1) - (1 : Int) := by\n by_induc\n \u00b7 -- Base Case\n rewrite [sum_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : Sum i from 0 to n, (2 : Int) ^ i =\n (2 : Int) ^ (n + 1) - (1 : Int)\n show Sum i from 0 to n + 1, (2 : Int) ^ i =\n (2 : Int) ^ (n + 1 + 1) - (1 : Int) from\n calc Sum i from 0 to n + 1, (2 : Int) ^ i\n _ = (Sum i from 0 to n, (2 : Int) ^ i)\n + (2 : Int) ^ (n + 1) := by rw [sum_from_zero_step]\n _ = (2 : Int) ^ (n + 1) - (1 : Int)\n + (2 : Int) ^ (n + 1) := by rw [ih]\n _ = (2 : Int) ^ (n + 1 + 1) - (1 : Int) := by ring\n done\n done\n\n/- Section 6.2 -/\nlemma Lemma_6_2_1_1 {A : Type} {R : BinRel A} {B : Set A} {b c : A}\n (h1 : partial_order R) (h2 : b \u2208 B) (h3 : minimalElt R c (B \\ {b}))\n (h4 : R b c) : minimalElt R b B := by\n define at h3\n --h3 : c \u2208 B \\ {b} \u2227 \u00ac\u2203 x \u2208 B \\ {b}, R x c \u2227 x \u2260 c\n define --Goal : b \u2208 B \u2227 \u00ac\u2203 x \u2208 B, R x b \u2227 x \u2260 b\n apply And.intro h2 --Goal : \u00ac\u2203 x \u2208 B, R x b \u2227 x \u2260 b\n contradict h3.right with h5\n obtain (x : A) (h6 : x \u2208 B \u2227 R x b \u2227 x \u2260 b) from h5\n apply Exists.intro x --Goal : x \u2208 B \\ {b} \u2227 R x c \u2227 x \u2260 c\n apply And.intro\n \u00b7 -- Proof that x \u2208 B \\ {b}\n show x \u2208 B \\ {b} from And.intro h6.left h6.right.right\n done\n \u00b7 -- Proof that R x c \u2227 x \u2260 c\n have Rtrans : transitive R := h1.right.left\n have h7 : R x c := Rtrans x b c h6.right.left h4\n apply And.intro h7\n by_contra h8\n rewrite [h8] at h6 --h6 : c \u2208 B \u2227 R c b \u2227 c \u2260 b\n have Rantisymm : antisymmetric R := h1.right.right\n have h9 : c = b := Rantisymm c b h6.right.left h4\n show False from h6.right.right h9\n done\n done\n\nlemma Lemma_6_2_1_2 {A : Type} {R : BinRel A} {B : Set A} {b c : A}\n (h1 : partial_order R) (h2 : b \u2208 B) (h3 : minimalElt R c (B \\ {b}))\n (h4 : \u00acR b c) : minimalElt R c B := sorry\n\ntheorem Example_6_2_1 {A : Type} (R : BinRel A) (h : partial_order R) :\n \u2200 n \u2265 1, \u2200 (B : Set A), numElts B n \u2192\n \u2203 (x : A), minimalElt R x B := by\n by_induc\n \u00b7 -- Base Case\n fix B : Set A\n assume h2 : numElts B 1\n rewrite [one_elt_iff_singleton] at h2\n obtain (b : A) (h3 : B = {b}) from h2\n apply Exists.intro b\n define --Goal : b \u2208 B \u2227 \u00ac\u2203 x \u2208 B, R x b \u2227 x \u2260 b\n apply And.intro\n \u00b7 -- Proof that b \u2208 B\n rewrite [h3] --Goal : b \u2208 {b}\n define --Goal : b = b\n rfl\n done\n \u00b7 -- Proof that nothing in B is smaller than b\n by_contra h4\n obtain (x : A) (h5 : x \u2208 B \u2227 R x b \u2227 x \u2260 b) from h4\n have h6 : x \u2208 B := h5.left\n rewrite [h3] at h6 --h6 : x \u2208 {b}\n define at h6 --h6 : x = b\n show False from h5.right.right h6\n done\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume h2 : n \u2265 1\n assume ih : \u2200 (B : Set A), numElts B n \u2192 \u2203 (x : A), minimalElt R x B\n fix B : Set A\n assume h3 : numElts B (n + 1)\n have h4 : n + 1 > 0 := by linarith\n obtain (b : A) (h5 : b \u2208 B) from nonempty_of_pos_numElts h3 h4\n set B' : Set A := B \\ {b}\n have h6 : numElts B' n := remove_one_numElts h3 h5\n obtain (c : A) (h7 : minimalElt R c B') from ih B' h6\n by_cases h8 : R b c\n \u00b7 -- Case 1. h8 : R b c\n have h9 : minimalElt R b B := Lemma_6_2_1_1 h h5 h7 h8\n show \u2203 (x : A), minimalElt R x B from Exists.intro b h9\n done\n \u00b7 -- Case 2. h8 : \u00acR b c\n have h9 : minimalElt R c B := Lemma_6_2_1_2 h h5 h7 h8\n show \u2203 (x : A), minimalElt R x B from Exists.intro c h9\n done\n done\n done\n\nlemma extendPO_is_ref {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : reflexive (extendPO R b) := sorry\n\nlemma extendPO_is_trans {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : transitive (extendPO R b) := sorry\n\nlemma extendPO_is_antisymm {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : antisymmetric (extendPO R b) := sorry\n\n", "theoremStatement": "lemma extendPO_is_po {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : partial_order (extendPO R b) ", "theoremName": "HTPI.extendPO_is_po", "fileCreated": {"commit": "cc48e6ba4c656bf74f7690323b2c27f2c466665b", "date": "2023-03-11"}, "theoremCreated": {"commit": "190e34021c4943fc63cee05edf567875671378bb", "date": "2023-02-18"}, "file": "htpi/HTPILib/Chap6.lean", "module": "HTPILib.Chap6", "jsonFile": "HTPILib.Chap6.jsonl", "positionMetadata": {"lineInFile": 240, "tokenPositionInFile": 7785, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 9, "numPremises": 11, "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", 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"Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n And.intro (extendPO_is_ref R b h)\n (And.intro (extendPO_is_trans R b h) (extendPO_is_antisymm R b h))", "proofType": "term", "proofLengthLines": 2, "proofLengthTokens": 109}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap8Part1\nnamespace HTPI\n\n/- Definitions and theorems in Chap8Part1 and HTPIDefs\n\ntheorem zero_elts_iff_empty {A : Type} (X : Set A) :\n numElts X 0 \u2194 empty X\n\ntheorem one_elt_iff_singleton {A : Type} (X : Set A) :\n numElts X 1 \u2194 \u2203 (x : A), X = {x}\n\ntheorem nonempty_of_pos_numElts {A : Type} {X : Set A} {n : Nat}\n (h1 : numElts X n) (h2 : n > 0) : \u2203 (x : A), x \u2208 X\n\ntheorem remove_one_numElts {A : Type} {X : Set A} {n : Nat} {x : A}\n (h1 : numElts X (n + 1)) (h2 : x \u2208 X) : numElts (X \\ {x}) n\n\ndef sum_seq {A : Type} [AddZeroClass A] (m k : Nat) (f : Nat \u2192 A) : A :=\n match m with\n | 0 => 0\n | n + 1 => sum_seq n k f + f (k + n)\n\ntheorem sum_base {A : Type} [AddZeroClass A] {k : Nat} {f : Nat \u2192 A} :\n Sum i from k to k, f i = f k\n\ntheorem sum_step {A : Type} [AddZeroClass A] {k n : Nat} {f : Nat \u2192 A}\n (h : k \u2264 n) : Sum i from k to (n + 1), f i = (Sum i from k to n, f i) + f (n + 1)\n\ntheorem sum_from_zero_step {A : Type} [AddZeroClass A] {n : Nat} {f : Nat \u2192 A} :\n Sum i from 0 to (n + 1), f i = (Sum i from 0 to n, f i) + f (n + 1)\n-/\n\n/- Definitions -/\ndef nat_even (n : Nat) : Prop := \u2203 (k : Nat), n = 2 * k\n\ndef nat_odd (n : Nat) : Prop := \u2203 (k : Nat), n = 2 * k + 1\n\ndef extendPO {A : Type} (R : BinRel A) (b : A) (x y : A) : Prop :=\n R x y \u2228 (R x b \u2227 \u00acR y b)\n\ndef fact (k : Nat) : Nat :=\n match k with\n | 0 => 1\n | n + 1 => (n + 1) * fact n\n\ndef Fib (n : Nat) : Nat :=\n match n with\n | 0 => 0\n | 1 => 1\n | k + 2 => Fib k + Fib (k + 1)\n\ndef rep_image {A : Type} (f : A \u2192 A) (n : Nat) (B : Set A) : Set A :=\n match n with\n | 0 => B\n | k + 1 => image f (rep_image f k B)\n\ndef cumul_image {A : Type} (f : A \u2192 A) (B : Set A) : Set A :=\n {x : A | \u2203 (n : Nat), x \u2208 rep_image f n B}\n\n/- Section 6.1 -/\ntheorem Like_Example_6_1_2 :\n \u2200 (n : Nat), 3 \u2223 n ^ 3 + 2 * n := by\n by_induc\n \u00b7 -- Base Case\n define --Goal : \u2203 (c : Nat), 0 ^ 3 + 2 * 0 = 3 * c\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : 3 \u2223 n ^ 3 + 2 * n\n define at ih --ih : \u2203 (c : Nat), n ^ 3 + 2 * n = 3 * c\n obtain (k : Nat) (h1 : n ^ 3 + 2 * n = 3 * k) from ih\n define --Goal : \u2203 (c : Nat), (n + 1) ^ 3 + 2 * (n + 1) = 3 * c\n apply Exists.intro (k + n ^ 2 + n + 1)\n show (n + 1) ^ 3 + 2 * (n + 1) = 3 * (k + n ^ 2 + n + 1) from\n calc (n + 1) ^ 3 + 2 * (n + 1)\n _ = n ^ 3 + 2 * n + 3 * n ^ 2 + 3 * n + 3 := by ring\n _ = 3 * k + 3 * n ^ 2 + 3 * n + 3 := by rw [h1]\n _ = 3 * (k + n ^ 2 + n + 1) := by ring\n done\n done\n\ntheorem Like_Example_6_1_1 :\n \u2200 (n : Nat), (Sum i from 0 to n, 2 ^ i) + 1 = 2 ^ (n + 1) := by\n by_induc\n \u00b7 -- Base Case\n rewrite [sum_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (Sum i from 0 to n, 2 ^ i) + 1 = 2 ^ (n + 1)\n show (Sum i from 0 to n + 1, 2 ^ i) + 1 = 2 ^ (n + 1 + 1) from\n calc (Sum i from 0 to n + 1, 2 ^ i) + 1\n _ = (Sum i from 0 to n, 2 ^ i) + 2 ^ (n + 1) + 1 := by\n rw [sum_from_zero_step]\n _ = (Sum i from 0 to n, 2 ^ i) + 1 + 2 ^ (n + 1) := by ring\n _ = 2 ^ (n + 1) + 2 ^ (n + 1) := by rw [ih]\n _ = 2 ^ (n + 1 + 1) := by ring\n done\n done\n\ntheorem Example_6_1_3 : \u2200 n \u2265 5, 2 ^ n > n ^ 2 := by\n by_induc\n \u00b7 -- Base Case\n decide\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume h1 : n \u2265 5\n assume ih : 2 ^ n > n ^ 2\n have h2 : n * n \u2265 5 * n := Nat.mul_le_mul_right n h1\n show 2 ^ (n + 1) > (n + 1) ^ 2 from\n calc 2 ^ (n + 1)\n _ = 2 * 2 ^ n := by ring\n _ > 2 * n ^ 2 := by linarith\n _ \u2265 n ^ 2 + 5 * n := by linarith\n _ > n ^ 2 + 2 * n + 1 := by linarith\n _ = (n + 1) ^ 2 := by ring\n done\n done\n\n#eval 2 - 3 --Answer: 0\n\ntheorem Example_6_1_1 :\n \u2200 (n : Nat), Sum i from 0 to n, (2 : Int) ^ i =\n (2 : Int) ^ (n + 1) - (1 : Int) := by\n by_induc\n \u00b7 -- Base Case\n rewrite [sum_base]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : Sum i from 0 to n, (2 : Int) ^ i =\n (2 : Int) ^ (n + 1) - (1 : Int)\n show Sum i from 0 to n + 1, (2 : Int) ^ i =\n (2 : Int) ^ (n + 1 + 1) - (1 : Int) from\n calc Sum i from 0 to n + 1, (2 : Int) ^ i\n _ = (Sum i from 0 to n, (2 : Int) ^ i)\n + (2 : Int) ^ (n + 1) := by rw [sum_from_zero_step]\n _ = (2 : Int) ^ (n + 1) - (1 : Int)\n + (2 : Int) ^ (n + 1) := by rw [ih]\n _ = (2 : Int) ^ (n + 1 + 1) - (1 : Int) := by ring\n done\n done\n\n/- Section 6.2 -/\nlemma Lemma_6_2_1_1 {A : Type} {R : BinRel A} {B : Set A} {b c : A}\n (h1 : partial_order R) (h2 : b \u2208 B) (h3 : minimalElt R c (B \\ {b}))\n (h4 : R b c) : minimalElt R b B := by\n define at h3\n --h3 : c \u2208 B \\ {b} \u2227 \u00ac\u2203 x \u2208 B \\ {b}, R x c \u2227 x \u2260 c\n define --Goal : b \u2208 B \u2227 \u00ac\u2203 x \u2208 B, R x b \u2227 x \u2260 b\n apply And.intro h2 --Goal : \u00ac\u2203 x \u2208 B, R x b \u2227 x \u2260 b\n contradict h3.right with h5\n obtain (x : A) (h6 : x \u2208 B \u2227 R x b \u2227 x \u2260 b) from h5\n apply Exists.intro x --Goal : x \u2208 B \\ {b} \u2227 R x c \u2227 x \u2260 c\n apply And.intro\n \u00b7 -- Proof that x \u2208 B \\ {b}\n show x \u2208 B \\ {b} from And.intro h6.left h6.right.right\n done\n \u00b7 -- Proof that R x c \u2227 x \u2260 c\n have Rtrans : transitive R := h1.right.left\n have h7 : R x c := Rtrans x b c h6.right.left h4\n apply And.intro h7\n by_contra h8\n rewrite [h8] at h6 --h6 : c \u2208 B \u2227 R c b \u2227 c \u2260 b\n have Rantisymm : antisymmetric R := h1.right.right\n have h9 : c = b := Rantisymm c b h6.right.left h4\n show False from h6.right.right h9\n done\n done\n\nlemma Lemma_6_2_1_2 {A : Type} {R : BinRel A} {B : Set A} {b c : A}\n (h1 : partial_order R) (h2 : b \u2208 B) (h3 : minimalElt R c (B \\ {b}))\n (h4 : \u00acR b c) : minimalElt R c B := sorry\n\ntheorem Example_6_2_1 {A : Type} (R : BinRel A) (h : partial_order R) :\n \u2200 n \u2265 1, \u2200 (B : Set A), numElts B n \u2192\n \u2203 (x : A), minimalElt R x B := by\n by_induc\n \u00b7 -- Base Case\n fix B : Set A\n assume h2 : numElts B 1\n rewrite [one_elt_iff_singleton] at h2\n obtain (b : A) (h3 : B = {b}) from h2\n apply Exists.intro b\n define --Goal : b \u2208 B \u2227 \u00ac\u2203 x \u2208 B, R x b \u2227 x \u2260 b\n apply And.intro\n \u00b7 -- Proof that b \u2208 B\n rewrite [h3] --Goal : b \u2208 {b}\n define --Goal : b = b\n rfl\n done\n \u00b7 -- Proof that nothing in B is smaller than b\n by_contra h4\n obtain (x : A) (h5 : x \u2208 B \u2227 R x b \u2227 x \u2260 b) from h4\n have h6 : x \u2208 B := h5.left\n rewrite [h3] at h6 --h6 : x \u2208 {b}\n define at h6 --h6 : x = b\n show False from h5.right.right h6\n done\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume h2 : n \u2265 1\n assume ih : \u2200 (B : Set A), numElts B n \u2192 \u2203 (x : A), minimalElt R x B\n fix B : Set A\n assume h3 : numElts B (n + 1)\n have h4 : n + 1 > 0 := by linarith\n obtain (b : A) (h5 : b \u2208 B) from nonempty_of_pos_numElts h3 h4\n set B' : Set A := B \\ {b}\n have h6 : numElts B' n := remove_one_numElts h3 h5\n obtain (c : A) (h7 : minimalElt R c B') from ih B' h6\n by_cases h8 : R b c\n \u00b7 -- Case 1. h8 : R b c\n have h9 : minimalElt R b B := Lemma_6_2_1_1 h h5 h7 h8\n show \u2203 (x : A), minimalElt R x B from Exists.intro b h9\n done\n \u00b7 -- Case 2. h8 : \u00acR b c\n have h9 : minimalElt R c B := Lemma_6_2_1_2 h h5 h7 h8\n show \u2203 (x : A), minimalElt R x B from Exists.intro c h9\n done\n done\n done\n\nlemma extendPO_is_ref {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : reflexive (extendPO R b) := sorry\n\nlemma extendPO_is_trans {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : transitive (extendPO R b) := sorry\n\nlemma extendPO_is_antisymm {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : antisymmetric (extendPO R b) := sorry\n\nlemma extendPO_is_po {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) : partial_order (extendPO R b) :=\n And.intro (extendPO_is_ref R b h)\n (And.intro (extendPO_is_trans R b h) (extendPO_is_antisymm R b h))\n\nlemma extendPO_extends {A : Type} (R : BinRel A) (b : A) (x y : A) :\n R x y \u2192 extendPO R b x y := by\n assume h1 : R x y\n define\n show R x y \u2228 R x b \u2227 \u00acR y b from Or.inl h1\n done\n\nlemma extendPO_all_comp {A : Type} (R : BinRel A) (b : A)\n (h : partial_order R) :\n \u2200 (x : A), extendPO R b b x \u2228 extendPO R b x b := by\n have Rref : reflexive R := h.left\n fix x : A\n or_left with h1\n define at h1 --h1 : \u00ac(R x b \u2228 R x b \u2227 \u00acR b b)\n demorgan at h1 --h1 : \u00acR x b \u2227 \u00ac(R x b \u2227 \u00acR b b)\n define --Goal : R b x \u2228 R b b \u2227 \u00acR x b\n apply Or.inr\n show R b b \u2227 \u00acR x b from And.intro (Rref b) h1.left\n done\n\ntheorem Exercise_6_2_2 {A : Type} (R : BinRel A) (h : partial_order R) :\n \u2200 (n : Nat) (B : Set A), numElts B n \u2192 \u2203 (T : BinRel A),\n partial_order T \u2227 (\u2200 (x y : A), R x y \u2192 T x y) \u2227\n \u2200 x \u2208 B, \u2200 (y : A), T x y \u2228 T y x := by\n by_induc\n \u00b7 -- Base Case\n fix B : Set A\n assume h2 : numElts B 0\n rewrite [zero_elts_iff_empty] at h2\n define at h2 --h2 : \u00ac\u2203 (x : A), x \u2208 B\n apply Exists.intro R\n apply And.intro h\n apply And.intro\n \u00b7 -- Proof that R extends R\n fix x : A; fix y : A\n assume h3 : R x y\n show R x y from h3\n done\n \u00b7 -- Proof that everything in B comparable to everything under R\n fix x : A\n assume h3 : x \u2208 B\n contradict h2\n show \u2203 (x : A), x \u2208 B from Exists.intro x h3\n done\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (B : Set A), numElts B n \u2192 \u2203 (T : BinRel A),\n partial_order T \u2227 (\u2200 (x y : A), R x y \u2192 T x y) \u2227\n \u2200 (x : A), x \u2208 B \u2192 \u2200 (y : A), T x y \u2228 T y x\n fix B : Set A\n assume h2 : numElts B (n + 1)\n have h3 : n + 1 > 0 := by linarith\n obtain (b : A) (h4 : b \u2208 B) from nonempty_of_pos_numElts h2 h3\n set B' : Set A := B \\ {b}\n have h5 : numElts B' n := remove_one_numElts h2 h4\n have h6 : \u2203 (T : BinRel A), partial_order T \u2227\n (\u2200 (x y : A), R x y \u2192 T x y) \u2227\n \u2200 (x : A), x \u2208 B' \u2192 \u2200 (y : A), T x y \u2228 T y x := ih B' h5\n obtain (T' : BinRel A)\n (h7 : partial_order T' \u2227 (\u2200 (x y : A), R x y \u2192 T' x y) \u2227\n \u2200 (x : A), x \u2208 B' \u2192 \u2200 (y : A), T' x y \u2228 T' y x) from h6\n have T'po : partial_order T' := h7.left\n have T'extR : \u2200 (x y : A), R x y \u2192 T' x y := h7.right.left\n have T'compB' : \u2200 (x : A), x \u2208 B' \u2192\n \u2200 (y : A), T' x y \u2228 T' y x := h7.right.right\n set T : BinRel A := extendPO T' b\n apply Exists.intro T\n apply And.intro (extendPO_is_po T' b T'po)\n apply And.intro\n \u00b7 -- Proof that T extends R\n fix x : A; fix y : A\n assume h8 : R x y\n have h9 : T' x y := T'extR x y h8\n show T x y from (extendPO_extends T' b x y h9)\n done\n \u00b7 -- Proof that everything in B comparable to everything under T\n fix x : A\n assume h8 : x \u2208 B\n by_cases h9 : x = b\n \u00b7 -- Case 1. h9 : x = b\n rewrite [h9]\n show \u2200 (y : A), T b y \u2228 T y b from extendPO_all_comp T' b T'po\n done\n \u00b7 -- Case 2. h9 : x \u2260 b\n have h10 : x \u2208 B' := And.intro h8 h9\n fix y : A\n have h11 : T' x y \u2228 T' y x := T'compB' x h10 y\n by_cases on h11\n \u00b7 -- Case 2.1. h11 : T' x y\n show T x y \u2228 T y x from\n Or.inl (extendPO_extends T' b x y h11)\n done\n \u00b7 -- Case 2.2. h11 : T' y x\n show T x y \u2228 T y x from\n Or.inr (extendPO_extends T' b y x h11)\n done\n done\n done\n done\n done\n\n/- Section 6.3 -/\n#eval fact 4 --Answer: 24\n\ntheorem Example_6_3_1 : \u2200 n \u2265 4, fact n > 2 ^ n := by\n by_induc\n \u00b7 -- Base Case\n decide\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume h1 : n \u2265 4\n assume ih : fact n > 2 ^ n\n have h2 : n + 1 > 2 := by linarith\n show fact (n + 1) > 2 ^ (n + 1) from\n calc fact (n + 1)\n _ = (n + 1) * fact n := by rfl\n _ > (n + 1) * 2 ^ n := by rel [ih]\n _ > 2 * 2 ^ n := by rel [h2]\n _ = 2 ^ (n + 1) := by ring\n done\n done\n\ntheorem Example_6_3_2_cheating : \u2200 (a : Real) (m n : Nat),\n a ^ (m + n) = a ^ m * a ^ n := by\n fix a : Real; fix m : Nat; fix n : Nat\n ring\n done\n\ntheorem Example_6_3_2 : \u2200 (a : Real) (m n : Nat),\n a ^ (m + n) = a ^ m * a ^ n := by\n fix a : Real; fix m : Nat\n --Goal : \u2200 (n : Nat), a ^ (m + n) = a ^ m * a ^ n\n by_induc\n \u00b7 -- Base Case\n show a ^ (m + 0) = a ^ m * a ^ 0 from\n calc a ^ (m + 0)\n _ = a ^ m := by rfl\n _ = a ^ m * 1 := (mul_one (a ^ m)).symm\n _ = a ^ m * a ^ 0 := by rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : a ^ (m + n) = a ^ m * a ^ n\n show a ^ (m + (n + 1)) = a ^ m * a ^ (n + 1) from\n calc a ^ (m + (n + 1))\n _ = a ^ ((m + n) + 1) := by rw [add_assoc]\n _ = a ^ (m + n) * a := by rfl\n _ = (a ^ m * a ^ n) * a := by rw [ih]\n _ = a ^ m * (a ^ n * a) := by rw [mul_assoc]\n _ = a ^ m * (a ^ (n + 1)) := by rfl\n done\n done\n\ntheorem Example_6_3_4 : \u2200 (x : Real), x > -1 \u2192\n \u2200 (n : Nat), (1 + x) ^ n \u2265 1 + n * x := by\n fix x : Real\n assume h1 : x > -1\n by_induc\n \u00b7 -- Base Case\n rewrite [Nat.cast_zero]\n linarith\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : (1 + x) ^ n \u2265 1 + n * x\n rewrite [Nat.cast_succ]\n have h2 : 0 \u2264 1 + x := by linarith\n have h3 : x ^ 2 \u2265 0 := sq_nonneg x\n have h4 : n * x ^ 2 \u2265 0 :=\n calc n * x ^ 2\n _ \u2265 n * 0 := by rel [h3]\n _ = 0 := by ring\n show (1 + x) ^ (n + 1) \u2265 1 + (n + 1) * x from\n calc (1 + x) ^ (n + 1)\n _ = (1 + x) ^ n * (1 + x) := by rfl\n _ \u2265 (1 + n * x) * (1 + x) := by rel [ih]\n _ = 1 + n * x + x + n * x ^ 2 := by ring\n _ \u2265 1 + n * x + x + 0 := by rel [h4]\n _ = 1 + (n + 1) * x := by ring\n done\n done\n\n/- Section 6.4 -/\ntheorem Example_6_4_1 : \u2200 m > 0, \u2200 (n : Nat),\n \u2203 (q r : Nat), n = m * q + r \u2227 r < m := by\n fix m : Nat\n assume h1 : m > 0\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, \u2203 (q r : Nat), n_1 = m * q + r \u2227 r < m\n by_cases h2 : n < m\n \u00b7 -- Case 1. h2 : n < m\n apply Exists.intro 0\n apply Exists.intro n --Goal : n = m * 0 + n \u2227 n < m\n apply And.intro _ h2\n ring\n done\n \u00b7 -- Case 2. h2 : \u00acn < m\n have h3 : m \u2264 n := by linarith\n obtain (k : Nat) (h4 : n = k + m) from Nat.exists_eq_add_of_le' h3\n have h5 : k < n := by linarith\n have h6 : \u2203 (q r : Nat), k = m * q + r \u2227 r < m := ih k h5\n obtain (q' : Nat)\n (h7 : \u2203 (r : Nat), k = m * q' + r \u2227 r < m) from h6\n obtain (r' : Nat) (h8 : k = m * q' + r' \u2227 r' < m) from h7\n apply Exists.intro (q' + 1)\n apply Exists.intro r' --Goal : n = m * (q' + 1) + r' \u2227 r' < m\n apply And.intro _ h8.right\n show n = m * (q' + 1) + r' from\n calc n\n _ = k + m := h4\n _ = m * q' + r' + m := by rw [h8.left]\n _ = m * (q' + 1) + r' := by ring\n done\n done\n\nlemma exists_eq_add_one_of_ne_zero {n : Nat}\n (h1 : n \u2260 0) : \u2203 (k : Nat), n = k + 1 := by\n have h2 : 1 \u2264 n := Nat.pos_of_ne_zero h1\n show \u2203 (k : Nat), n = k + 1 from Nat.exists_eq_add_of_le' h2\n done\n\ntheorem exists_eq_add_two_of_ne_zero_one {n : Nat}\n (h1 : n \u2260 0) (h2 : n \u2260 1) : \u2203 (k : Nat), n = k + 2 := by\n have h3 : 1 \u2264 n := Nat.pos_of_ne_zero h1\n have h4 : 2 \u2264 n := lt_of_le_of_ne' h3 h2\n show \u2203 (k : Nat), n = k + 2 from Nat.exists_eq_add_of_le' h4\n done\n\nexample : \u2200 (n : Nat), Fib n < 2 ^ n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, Fib n_1 < 2 ^ n_1\n by_cases h1 : n = 0\n \u00b7 -- Case 1. h1 : n = 0\n rewrite [h1] --Goal : Fib 0 < 2 ^ 0\n decide\n done\n \u00b7 -- Case 2. h1 : \u00acn = 0\n by_cases h2 : n = 1\n \u00b7 -- Case 2.1. h2 : n = 1\n rewrite [h2]\n decide\n done\n \u00b7 -- Case 2.2. h2 : \u00acn = 1\n obtain (k : Nat) (h3 : n = k + 2) from\n exists_eq_add_two_of_ne_zero_one h1 h2\n have h4 : k < n := by linarith\n have h5 : Fib k < 2 ^ k := ih k h4\n have h6 : k + 1 < n := by linarith\n have h7 : Fib (k + 1) < 2 ^ (k + 1) := ih (k + 1) h6\n rewrite [h3] --Goal : Fib (k + 2) < 2 ^ (k + 2)\n show Fib (k + 2) < 2 ^ (k + 2) from\n calc Fib (k + 2)\n _ = Fib k + Fib (k + 1) := by rfl\n _ < 2 ^ k + Fib (k + 1) := by rel [h5]\n _ < 2 ^ k + 2 ^ (k + 1) := by rel [h7]\n _ \u2264 2 ^ k + 2 ^ (k + 1) + 2 ^ k := by linarith\n _ = 2 ^ (k + 2) := by ring\n done\n done\n done\n\ntheorem well_ord_princ (S : Set Nat) : (\u2203 (n : Nat), n \u2208 S) \u2192\n \u2203 n \u2208 S, \u2200 m \u2208 S, n \u2264 m := by\n contrapos\n assume h1 : \u00ac\u2203 n \u2208 S, \u2200 m \u2208 S, n \u2264 m\n quant_neg --Goal : \u2200 (n : Nat), n \u2209 S\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2209 S --Goal : n \u2209 S\n contradict h1 with h2 --h2 : n \u2208 S\n --Goal : \u2203 n \u2208 S, \u2200 m \u2208 S, n \u2264 m\n apply Exists.intro n --Goal : n \u2208 S \u2227 \u2200 m \u2208 S, n \u2264 m\n apply And.intro h2 --Goal : \u2200 m \u2208 S, n \u2264 m\n fix m : Nat\n assume h3 : m \u2208 S\n have h4 : m < n \u2192 m \u2209 S := ih m\n contrapos at h4 --h4 : m \u2208 S \u2192 \u00acm < n\n have h5 : \u00acm < n := h4 h3\n linarith\n done\n\nlemma sq_even_iff_even (n : Nat) : nat_even (n * n) \u2194 nat_even n := sorry\n\ntheorem Theorem_6_4_5 :\n \u00ac\u2203 (q p : Nat), p * p = 2 * (q * q) \u2227 q \u2260 0 := by\n set S : Set Nat :=\n {q : Nat | \u2203 (p : Nat), p * p = 2 * (q * q) \u2227 q \u2260 0}\n by_contra h1\n have h2 : \u2203 (q : Nat), q \u2208 S := h1\n have h3 : \u2203 q \u2208 S, \u2200 r \u2208 S, q \u2264 r := well_ord_princ S h2\n obtain (q : Nat) (h4 : q \u2208 S \u2227 \u2200 r \u2208 S, q \u2264 r) from h3\n have qinS : q \u2208 S := h4.left\n have qleast : \u2200 r \u2208 S, q \u2264 r := h4.right\n define at qinS --qinS : \u2203 (p : Nat), p * p = 2 * (q * q) \u2227 q \u2260 0\n obtain (p : Nat) (h5 : p * p = 2 * (q * q) \u2227 q \u2260 0) from qinS\n have pqsqrt2 : p * p = 2 * (q * q) := h5.left\n have qne0 : q \u2260 0 := h5.right\n have h6 : nat_even (p * p) := Exists.intro (q * q) pqsqrt2\n rewrite [sq_even_iff_even p] at h6 --h6 : nat_even p\n obtain (p' : Nat) (p'halfp : p = 2 * p') from h6\n have h7 : 2 * (2 * (p' * p')) = 2 * (q * q) := by\n rewrite [\u2190pqsqrt2, p'halfp]\n ring\n done\n have h8 : 2 > 0 := by decide\n rewrite [mul_left_cancel_iff_of_pos h8] at h7\n --h7 : 2 * (p' * p') = q * q\n have h9 : nat_even (q * q) := Exists.intro (p' * p') h7.symm\n rewrite [sq_even_iff_even q] at h9 --h9 : nat_even q\n obtain (q' : Nat) (q'halfq : q = 2 * q') from h9\n have h10 : 2 * (p' * p') = 2 * (2 * (q' * q')) := by\n rewrite [h7, q'halfq]\n ring\n done\n rewrite [mul_left_cancel_iff_of_pos h8] at h10\n --h10 : p' * p' = 2 * (q' * q')\n have q'ne0 : q' \u2260 0 := by\n contradict qne0 with h11\n rewrite [q'halfq, h11]\n rfl\n done\n have q'inS : q' \u2208 S := Exists.intro p' (And.intro h10 q'ne0)\n have qleq' : q \u2264 q' := qleast q' q'inS\n rewrite [q'halfq] at qleq' --qleq' : 2 * q' \u2264 q'\n contradict q'ne0\n linarith\n done\n\n/- Section 6.5 -/\ntheorem rep_image_base {A : Type} (f : A \u2192 A) (B : Set A) :\n rep_image f 0 B = B := by rfl\n\ntheorem rep_image_step {A : Type} (f : A \u2192 A) (n : Nat) (B : Set A) :\n rep_image f (n + 1) B = image f (rep_image f n B) := by rfl\n\n", "theoremStatement": "lemma rep_image_sub_closed {A : Type} {f : A \u2192 A} {B D : Set A}\n (h1 : B \u2286 D) (h2 : closed f D) :\n \u2200 (n : Nat), rep_image f n B \u2286 D ", "theoremName": "HTPI.rep_image_sub_closed", "fileCreated": {"commit": "cc48e6ba4c656bf74f7690323b2c27f2c466665b", "date": "2023-03-11"}, "theoremCreated": {"commit": "190e34021c4943fc63cee05edf567875671378bb", "date": "2023-02-18"}, "file": 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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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n by_induc\n \u00b7 -- Base Case\n rewrite [rep_image_base] --Goal : B \u2286 D\n show B \u2286 D from h1\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : rep_image f n B \u2286 D --Goal : rep_image f (n + 1) B \u2286 D\n fix x : A\n assume h3 : x \u2208 rep_image f (n + 1) B --Goal : x \u2208 D\n rewrite [rep_image_step] at h3\n define at h3 --h3 : \u2203 x_1 \u2208 rep_image f n B, f x_1 = x\n obtain (b : A) (h4 : b \u2208 rep_image f n B \u2227 f b = x) from h3\n rewrite [\u2190h4.right] --Goal : f b \u2208 D\n have h5 : b \u2208 D := ih h4.left\n define at h2 --h2 : \u2200 x \u2208 D, f x \u2208 D\n show f b \u2208 D from h2 b h5\n done\n done", "proofType": "tactic", "proofLengthLines": 19, "proofLengthTokens": 635}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap7\nnamespace HTPI\n\n/- Definitions -/\ndef fnz (n : Nat) : Int := if 2 \u2223 n then \u2191(n / 2) else -\u2191((n + 1) / 2)\n\ndef fzn (a : Int) : Nat := if a \u2265 0 then 2 * Int.toNat a else 2 * Int.toNat (-a) - 1\n\ndef tri (k : Nat) : Nat := k * (k + 1) / 2\n\ndef fnnn (p : Nat \u00d7 Nat) : Nat := tri (p.1 + p.2) + p.1\n\ndef num_elts_below (A : Set Nat) (m s : Nat) : Prop :=\n match m with\n | 0 => s = 0\n | n + 1 => (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s)\n\ndef enum (A : Set Nat) (s n : Nat) : Prop := n \u2208 A \u2227 num_elts_below A n s\n\ndef unique_val_on_N {U : Type} (R : Rel Nat U) : Prop :=\n \u2200 \u2983n : Nat\u2984 \u2983x1 x2 : U\u2984, R n x1 \u2192 R n x2 \u2192 x1 = x2\n\ndef nat_rel_onto {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203 (n : Nat), R n x\n\ndef fcnl_onto_from_nat {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n unique_val_on_N R \u2227 nat_rel_onto R A\n\ndef fcnl_one_one_to_nat {U : Type} (R : Rel U Nat) (A : Set U) : Prop :=\n fcnl_on R A \u2227 \u2200 \u2983x1 x2 : U\u2984 \u2983n : Nat\u2984,\n (x1 \u2208 A \u2227 R x1 n) \u2192 (x2 \u2208 A \u2227 R x2 n) \u2192 x1 = x2\n\ndef least_rel_to {U : Type} (S : Rel Nat U) (x : U) (n : Nat) : Prop :=\n S n x \u2227 \u2200 (m : Nat), S m x \u2192 n \u2264 m\n\ndef restrict_to {U V : Type} (S : Rel U V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 S x y\n\ndef fqn (q : Rat) : Nat := fnnn (fzn q.num, q.den)\n\ndef Set_rp_below (m : Nat) : Set Nat := {n : Nat | rel_prime m n \u2227 n < m}\n\ndef Set_prod {U V : Type} (A : Set U) (B : Set V) : Set (U \u00d7 V) :=\n {(a, b) : U \u00d7 V | a \u2208 A \u2227 b \u2208 B}\n\nnotation:75 A:75 \" \u00d7\u209b \" B:75 => Set_prod A B\n\ndef Rel_prod {U V W X : Type} (R : Rel U V) (S : Rel W X)\n (p : U \u00d7 W) (q : V \u00d7 X) : Prop := R p.1 q.1 \u2227 S p.2 q.2\n\nnotation:75 R:75 \" \u00d7\u1d63 \" S:75 => Rel_prod R S\n\ndef qr (n a : Nat) : Nat \u00d7 Nat := (a / n, a % n)\n\ndef mod_mod (m n a : Nat) : Nat \u00d7 Nat := (a % m, a % n)\n\ndef enum_union_fam {U : Type}\n (F : Set (Set U)) (f : Set U \u2192 Rel Nat U) (R : Rel Nat (Set U))\n (n : Nat) (a : U) : Prop := \u2203 (p : Nat \u00d7 Nat), fnnn p = n \u2227\n \u2203 A \u2208 F, R p.1 A \u2227 (f A) p.2 a\n\ndef seq {U : Type} (A : Set U) : Set (List U) :=\n {l : List U | \u2200 x \u2208 l, x \u2208 A}\n\ndef seq_by_length {U : Type} (A : Set U) (n : Nat) : Set (List U) :=\n {l : List U | l \u2208 seq A \u2227 l.length = n}\n\ndef seq_cons (U : Type) (p : U \u00d7 (List U)) : List U := p.1 :: p.2\n\ndef sbl_set {U : Type} (A : Set U) : Set (Set (List U)) :=\n {S : Set (List U) | \u2203 (n : Nat), seq_by_length A n = S}\n\ndef rep_common_image\n {U V : Type} (R S : Rel U V) (X0 : Set U) (n : Nat) : Set U :=\n match n with\n | 0 => X0\n | m + 1 => {a : U | \u2203 x \u2208 rep_common_image R S X0 m,\n \u2203 (y : V), R x y \u2227 S a y}\n\ndef cum_rep_image {U V : Type} (R S : Rel U V) (X0 : Set U) : Set U :=\n {a : U | \u2203 (n : Nat), a \u2208 rep_common_image R S X0 n}\n\ndef csb_match {U V : Type} (R S : Rel U V) (X0 : Set U)\n (x : U) (y : V) : Prop := x \u2208 cum_rep_image R S X0 \u2227 R x y \u2228\n x \u2209 cum_rep_image R S X0 \u2227 S x y\n\n/- Section 8.1 -/\n#eval [fnz 0, fnz 1, fnz 2, fnz 3, fnz 4, fnz 5, fnz 6]\n --Answer: [0, -1, 1, -2, 2, -3, 3]\n\n#eval [fzn 0, fzn (-1), fzn 1, fzn (-2), fzn 2, fzn (-3), fzn 3]\n --Answer: [0, 1, 2, 3, 4, 5, 6]\n\nlemma fnz_even (k : Nat) : fnz (2 * k) = \u2191k := by\n have h1 : 2 \u2223 2 * k := by\n apply Exists.intro k\n rfl\n done\n have h2 : fnz (2 * k) = if 2 \u2223 2 * k then \u2191(2 * k / 2)\n else -\u2191((2 * k + 1) / 2) := by rfl\n rewrite [if_pos h1] at h2 --h2 : fnz (2 * k) = \u2191(2 * k / 2)\n have h3 : 0 < 2 := by linarith\n rewrite [Nat.mul_div_cancel_left k h3] at h2\n show fnz (2 * k) = \u2191k from h2\n done\n\nlemma fnz_odd (k : Nat) : fnz (2 * k + 1) = -\u2191(k + 1) := sorry\n\nlemma fzn_nat (k : Nat) : fzn \u2191k = 2 * k := by rfl\n\nlemma fzn_neg_succ_nat (k : Nat) : fzn (-\u2191(k + 1)) = 2 * k + 1 := by rfl\n\n--From exercises of Section 6.1\ntheorem Exercise_6_1_16a1 : \u2200 (n : Nat), nat_even n \u2228 nat_odd n := sorry\n\nlemma fzn_fnz : fzn \u2218 fnz = id := by\n apply funext --Goal : \u2200 (x : Nat), (fzn \u2218 fnz) x = id x\n fix n : Nat\n rewrite [comp_def] --Goal : fzn (fnz n) = id n\n have h1 : nat_even n \u2228 nat_odd n := Exercise_6_1_16a1 n\n by_cases on h1\n \u00b7 -- Case 1. h1 : nat_even n\n obtain (k : Nat) (h2 : n = 2 * k) from h1\n rewrite [h2, fnz_even, fzn_nat]\n rfl\n done\n \u00b7 -- Case 2. h1 : nat_odd n\n obtain (k : Nat) (h2 : n = 2 * k + 1) from h1\n rewrite [h2, fnz_odd, fzn_neg_succ_nat]\n rfl\n done\n done\n\nlemma fnz_fzn : fnz \u2218 fzn = id := sorry\n\nlemma fzn_one_one : one_to_one fzn := Theorem_5_3_3_1 fzn fnz fnz_fzn\n\nlemma fzn_onto : onto fzn := Theorem_5_3_3_2 fzn fnz fzn_fnz\n\nlemma fnz_one_one : one_to_one fnz := Theorem_5_3_3_1 fnz fzn fzn_fnz\n\nlemma fnz_onto : onto fnz := Theorem_5_3_3_2 fnz fzn fnz_fzn\n\nlemma fnnn_def (a b : Nat) : fnnn (a, b) = tri (a + b) + a := by rfl\n\n#eval [fnnn (0, 0), fnnn (0, 1), fnnn (1, 0), fnnn (0, 2), fnnn (1, 1)]\n --Answer: [0, 1, 2, 3, 4]\n\nlemma tri_step (k : Nat) : tri (k + 1) = tri k + k + 1 := sorry\n\nlemma tri_incr {j k : Nat} (h1 : j \u2264 k) : tri j \u2264 tri k := sorry\n\nlemma le_of_fnnn_eq {a1 b1 a2 b2 : Nat}\n (h1 : fnnn (a1, b1) = fnnn (a2, b2)) : a1 + b1 \u2264 a2 + b2 := by\n by_contra h2\n have h3 : a2 + b2 + 1 \u2264 a1 + b1 := by linarith\n have h4 : fnnn (a2, b2) < fnnn (a1, b1) :=\n calc fnnn (a2, b2)\n _ = tri (a2 + b2) + a2 := by rfl\n _ < tri (a2 + b2) + (a2 + b2) + 1 := by linarith\n _ = tri (a2 + b2 + 1) := (tri_step _).symm\n _ \u2264 tri (a1 + b1) := tri_incr h3\n _ \u2264 tri (a1 + b1) + a1 := by linarith\n _ = fnnn (a1, b1) := by rfl\n linarith\n done\n\nlemma fnnn_one_one : one_to_one fnnn := by\n fix (a1, b1) : Nat \u00d7 Nat\n fix (a2, b2) : Nat \u00d7 Nat\n assume h1 : fnnn (a1, b1) = fnnn (a2, b2) --Goal : (a1, b1) = (a2, b2)\n have h2 : a1 + b1 \u2264 a2 + b2 := le_of_fnnn_eq h1\n have h3 : a2 + b2 \u2264 a1 + b1 := le_of_fnnn_eq h1.symm\n have h4 : a1 + b1 = a2 + b2 := by linarith\n rewrite [fnnn_def, fnnn_def, h4] at h1\n --h1 : tri (a2 + b2) + a1 = tri (a2 + b2) + a2\n have h6 : a1 = a2 := Nat.add_left_cancel h1\n rewrite [h6] at h4 --h4 : a2 + b1 = a2 + b2\n have h7 : b1 = b2 := Nat.add_left_cancel h4\n rewrite [h6, h7]\n rfl\n done\n\nlemma fnnn_onto : onto fnnn := by\n define --Goal : \u2200 (y : Nat), \u2203 (x : Nat \u00d7 Nat), fnnn x = y\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro (0, 0)\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2203 (x : Nat \u00d7 Nat), fnnn x = n\n obtain ((a, b) : Nat \u00d7 Nat) (h1 : fnnn (a, b) = n) from ih\n by_cases h2 : b = 0\n \u00b7 -- Case 1. h2 : b = 0\n apply Exists.intro (0, a + 1)\n show fnnn (0, a + 1) = n + 1 from\n calc fnnn (0, a + 1)\n _ = tri (0 + (a + 1)) + 0 := by rfl\n _ = tri (a + 1) := by ring\n _ = tri a + a + 1 := tri_step a\n _ = tri (a + 0) + a + 1 := by ring\n _ = fnnn (a, b) + 1 := by rw [h2, fnnn_def]\n _ = n + 1 := by rw [h1]\n done\n \u00b7 -- Case 2. h2 : b \u2260 0\n obtain (k : Nat) (h3 : b = k + 1) from\n exists_eq_add_one_of_ne_zero h2\n apply Exists.intro (a + 1, k)\n show fnnn (a + 1, k) = n + 1 from\n calc fnnn (a + 1, k)\n _ = tri (a + 1 + k) + (a + 1) := by rfl\n _ = tri (a + (k + 1)) + a + 1 := by ring\n _ = tri (a + b) + a + 1 := by rw [h3]\n _ = fnnn (a, b) + 1 := by rfl\n _ = n + 1 := by rw [h1]\n done\n done\n done\n\nlemma one_one_on_of_one_one {U V : Type} {f : U \u2192 V}\n (h : one_to_one f) (A : Set U) : one_one_on f A := by\n define\n fix x1 : U; fix x2 : U\n assume h1 : x1 \u2208 A\n assume h2 : x2 \u2208 A\n show f x1 = f x2 \u2192 x1 = x2 from h x1 x2\n done\n\nlemma elt_Univ {U : Type} (u : U) :\n u \u2208 Univ U := by trivial\n\ntheorem equinum_Univ {U V : Type} {f : U \u2192 V}\n (h1 : one_to_one f) (h2 : onto f) : Univ U \u223c Univ V := by\n have h3 : image f (Univ U) = Univ V := by\n apply Set.ext\n fix v : V\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h3 : v \u2208 image f (Univ U)\n show v \u2208 Univ V from elt_Univ v\n done\n \u00b7 -- (\u2190)\n assume h3 : v \u2208 Univ V\n obtain (u : U) (h4 : f u = v) from h2 v\n apply Exists.intro u\n apply And.intro _ h4\n show u \u2208 Univ U from elt_Univ u\n done\n done\n show Univ U \u223c Univ V from\n equinum_image (one_one_on_of_one_one h1 (Univ U)) h3\n done\n\ntheorem Z_equinum_N : Univ Int \u223c Univ Nat :=\n equinum_Univ fzn_one_one fzn_onto\n\ntheorem NxN_equinum_N : Univ (Nat \u00d7 Nat) \u223c Univ Nat :=\n equinum_Univ fnnn_one_one fnnn_onto\n\nlemma neb_step (A : Set Nat) (n s : Nat) : num_elts_below A (n + 1) s \u2194\n (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s) := by rfl\n\nlemma neb_step_elt {A : Set Nat} {n : Nat} (h1 : n \u2208 A) (s : Nat) :\n num_elts_below A (n + 1) s \u2194 1 \u2264 s \u2227 num_elts_below A n (s - 1) := by\n rewrite [neb_step]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) \u2228\n \u00acn \u2208 A \u2227 num_elts_below A n s\n by_cases on h2\n \u00b7 -- Case 1. h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n show 1 \u2264 s \u2227 num_elts_below A n (s - 1) from h2.right\n done\n \u00b7 -- Case 2. h2 : \u00acn \u2208 A \u2227 num_elts_below A n s\n show 1 \u2264 s \u2227 num_elts_below A n (s - 1) from absurd h1 h2.left\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n apply Or.inl\n show n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) from And.intro h1 h2\n done\n done\n\nlemma neb_step_not_elt {A : Set Nat} {n : Nat} (h1 : n \u2209 A) (s : Nat) :\n num_elts_below A (n + 1) s \u2194 num_elts_below A n s := by\n rewrite [neb_step]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1) \u2228\n \u00acn \u2208 A \u2227 num_elts_below A n s\n by_cases on h2\n \u00b7 -- Case 1. h2 : n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)\n show num_elts_below A n s from absurd h2.left h1\n done\n \u00b7 -- Case 2. h2 : \u00acn \u2208 A \u2227 num_elts_below A n s\n show num_elts_below A n s from h2.right\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : num_elts_below A n s\n apply Or.inr\n show \u00acn \u2208 A \u2227 num_elts_below A n s from And.intro h1 h2\n done\n done\n\nlemma neb_exists (A : Set Nat) :\n \u2200 (n : Nat), \u2203 (s : Nat), num_elts_below A n s := by\n by_induc\n \u00b7 -- Base Case\n apply Exists.intro 0\n define\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2203 (s : Nat), num_elts_below A n s\n obtain (t : Nat) (h1 : num_elts_below A n t) from ih\n by_cases h2 : n \u2208 A\n \u00b7 -- Case 1. h2 : n \u2208 A\n apply Exists.intro (t + 1)\n rewrite [neb_step_elt h2, Nat.add_sub_cancel]\n apply And.intro _ h1\n linarith\n done\n \u00b7 -- Case 2. h2 : n \u2209 A\n apply Exists.intro t\n rewrite [neb_step_not_elt h2]\n show num_elts_below A n t from h1\n done\n done\n done\n\nlemma neb_unique (A : Set Nat) : \u2200 \u2983n : Nat\u2984, \u2200 \u2983s1 s2 : Nat\u2984,\n num_elts_below A n s1 \u2192 num_elts_below A n s2 \u2192 s1 = s2 := by\n by_induc\n \u00b7 -- Base Case\n fix s1 : Nat; fix s2 : Nat\n assume h1 : num_elts_below A 0 s1\n assume h2 : num_elts_below A 0 s2\n define at h1; define at h2 --h1 : s1 = 0; h2 : s2 = 0\n rewrite [h1, h2]\n rfl\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 \u2983s1 s2 : Nat\u2984,\n num_elts_below A n s1 \u2192 num_elts_below A n s2 \u2192 s1 = s2\n fix s1 : Nat; fix s2 : Nat\n assume h1 : num_elts_below A (n + 1) s1\n assume h2 : num_elts_below A (n + 1) s2\n by_cases h3 : n \u2208 A\n \u00b7 -- Case 1. h3 : n \u2208 A\n rewrite [neb_step_elt h3] at h1\n rewrite [neb_step_elt h3] at h2\n --h1 : 1 \u2264 s1 \u2227 num_elts_below A n (s1 - 1)\n --h2 : 1 \u2264 s2 \u2227 num_elts_below A n (s2 - 1)\n have h4 : s1 - 1 = s2 - 1 := ih h1.right h2.right\n show s1 = s2 from\n calc s1\n _ = s1 - 1 + 1 := (Nat.sub_add_cancel h1.left).symm\n _ = s2 - 1 + 1 := by rw [h4]\n _ = s2 := Nat.sub_add_cancel h2.left\n done\n \u00b7 -- Case 2. h3 : n \u2209 A\n rewrite [neb_step_not_elt h3] at h1 --h1 : num_elts_below A n s1\n rewrite [neb_step_not_elt h3] at h2 --h2 : num_elts_below A n s2\n show s1 = s2 from ih h1 h2\n done\n done\n done\n\nlemma neb_increase {A : Set Nat} {n s : Nat} (h1 : enum A s n) :\n \u2200 \u2983m : Nat\u2984, m \u2265 n + 1 \u2192 \u2200 \u2983t : Nat\u2984, num_elts_below A m t \u2192 s < t := by\n by_induc\n \u00b7 -- Base Case\n define at h1\n fix t : Nat\n assume h2 : num_elts_below A (n + 1) t\n rewrite [neb_step_elt h1.left] at h2\n have h3 : s = t - 1 := neb_unique A h1.right h2.right\n show s < t from\n calc s\n _ = t - 1 := h3\n _ < t - 1 + 1 := by linarith\n _ = t := Nat.sub_add_cancel h2.left\n done\n \u00b7 -- Induction Step\n fix m : Nat\n assume h2 : m \u2265 n + 1\n assume ih : \u2200 \u2983t : Nat\u2984, num_elts_below A m t \u2192 s < t\n fix t : Nat\n assume h3 : num_elts_below A (m + 1) t\n by_cases h4 : m \u2208 A\n \u00b7 -- Case 1. h4 : m \u2208 A\n rewrite [neb_step_elt h4] at h3\n have h5 : s < t - 1 := ih h3.right\n show s < t from\n calc s\n _ < t - 1 := h5\n _ \u2264 t := Nat.sub_le _ _\n done\n \u00b7 -- Case 2. h4 : m \u2209 A\n rewrite [neb_step_not_elt h4] at h3\n show s < t from ih h3\n done\n done\n done\n\nlemma enum_not_skip {A : Set Nat} : \u2200 \u2983m s : Nat\u2984, num_elts_below A m s \u2192\n \u2200 t < s, \u2203 (n : Nat), enum A t n := by\n by_induc\n \u00b7 -- Base Case\n fix s : Nat\n assume h1 : num_elts_below A 0 s\n define at h1\n fix t : Nat\n contrapos\n assume h2 : \u00ac\u2203 (n : Nat), enum A t n\n linarith\n done\n \u00b7 -- Induction Step\n fix m : Nat\n assume ih : \u2200 \u2983s : Nat\u2984, num_elts_below A m s \u2192 \u2200 t < s, \u2203 (n : Nat), enum A t n\n fix s : Nat\n assume h1 : num_elts_below A (m + 1) s\n by_cases h2 : m \u2208 A\n \u00b7 -- Case 1. h2 : m \u2208 A\n rewrite [neb_step_elt h2] at h1\n have h3 : \u2200 t < s - 1, \u2203 (n : Nat), enum A t n := ih h1.right\n fix t : Nat\n assume h4 : t < s\n by_cases h5 : t = s - 1\n \u00b7 -- Case 1.1. h5 : t = s - 1\n apply Exists.intro m\n define\n apply And.intro h2\n rewrite [h5]\n show num_elts_below A m (s - 1) from h1.right\n done\n \u00b7 -- Case 1.2. h5 : t \u2260 s - 1\n have h6 : t \u2264 s - 1 := Nat.le_pred_of_lt h4\n have h7 : t < s - 1 := Nat.lt_of_le_of_ne h6 h5\n show \u2203 (n : Nat), enum A t n from ih h1.right t h7\n done\n done\n \u00b7 -- Case 2. h2 : m \u2209 A\n rewrite [neb_step_not_elt h2] at h1\n show \u2200 t < s, \u2203 (n : Nat), enum A t n from ih h1\n done\n done\n done\n\nlemma enum_le {A : Set Nat} {t n1 n2 : Nat}\n (h1 : enum A t n1) (h2 : enum A t n2) : n1 \u2264 n2 := by\n by_contra h3\n have h4 : n2 + 1 \u2264 n1 := by linarith\n define at h1\n have h5 : t < t := neb_increase h2 h4 h1.right\n linarith\n done\n\nlemma enum_unique (A : Set Nat) (t : Nat) :\n \u2200 \u2983n1 n2 : Nat\u2984, enum A t n1 \u2192 enum A t n2 \u2192 n1 = n2 := by\n fix n1 : Nat; fix n2 : Nat\n assume h1 : enum A t n1\n assume h2 : enum A t n2\n have h3 : n1 \u2264 n2 := enum_le h1 h2\n have h4 : n2 \u2264 n1 := enum_le h2 h1\n linarith\n done\n\nlemma inv_enum_fcnl (A : Set Nat) : fcnl_on (invRel (enum A)) A := by\n define\n fix n : Nat\n assume h1 : n \u2208 A\n exists_unique\n \u00b7 -- Existence\n obtain (s : Nat) (h2 : num_elts_below A n s) from neb_exists A n\n apply Exists.intro s\n define\n show n \u2208 A \u2227 num_elts_below A n s from And.intro h1 h2\n done\n \u00b7 -- Uniqueness\n fix s1 : Nat; fix s2 : Nat\n assume h2 : invRel (enum A) n s1\n assume h3 : invRel (enum A) n s2\n define at h2; define at h3\n show s1 = s2 from neb_unique A h2.right h3.right\n done\n done\n\nlemma bdd_subset_nat_match {A : Set Nat} {m s : Nat}\n (h1 : \u2200 n \u2208 A, n < m) (h2 : num_elts_below A m s) :\n matching (enum A) (I s) A := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix t : Nat; fix n : Nat\n assume h3 : enum A t n\n have h4 : \u2200 \u2983m : Nat\u2984, m \u2265 n + 1 \u2192 \u2200 \u2983t_1 : Nat\u2984, num_elts_below A m t_1 \u2192 t < t_1 :=\n neb_increase h3\n define at h3\n apply And.intro _ h3.left\n define\n have h5 : n < m := h1 n h3.left\n have h6 : m \u2265 n + 1 := h5\n show t < s from h4 h6 h2\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- proof of fcnl_on (enum A) (I s)\n define\n fix t : Nat\n assume h3 : t \u2208 I s\n define at h3\n exists_unique\n \u00b7 -- Existence\n show \u2203 (y : Nat), enum A t y from enum_not_skip h2 t h3\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A t y_1 \u2192 enum A t y_2 \u2192 y_1 = y_2 from\n enum_unique A t\n done\n done\n \u00b7 -- Proof of fcnl_on (invRel (enum A)) A\n show fcnl_on (invRel (enum A)) A from inv_enum_fcnl A\n done\n done\n done\n\nlemma bdd_subset_nat {A : Set Nat} {m s : Nat}\n (h1 : \u2200 n \u2208 A, n < m) (h2 : num_elts_below A m s) :\n I s \u223c A := Exists.intro (enum A) (bdd_subset_nat_match h1 h2)\n\nlemma enum_fcnl_of_unbdd {A : Set Nat} (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n fcnl_on (enum A) (Univ Nat) := by\n define\n by_induc\n \u00b7 -- Base Case\n assume h2 : 0 \u2208 Univ Nat\n exists_unique\n \u00b7 -- Existence\n obtain (n : Nat) (h3 : n \u2208 A \u2227 n \u2265 0) from h1 0\n obtain (s : Nat) (h4 : num_elts_below A (n + 1) s) from neb_exists A (n + 1)\n have h5 : \u2200 (t : Nat), t < s \u2192 \u2203 (n : Nat), enum A t n := enum_not_skip h4\n rewrite [neb_step_elt h3.left] at h4\n show \u2203 (y : Nat), enum A 0 y from h5 0 h4.left\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A 0 y_1 \u2192 enum A 0 y_2 \u2192 y_1 = y_2 from enum_unique A 0\n done\n done\n \u00b7 -- Induction Step\n fix s : Nat\n assume ih : s \u2208 Univ Nat \u2192 \u2203! (y : Nat), enum A s y\n assume h2 : s + 1 \u2208 Univ Nat\n exists_unique\n \u00b7 -- Existence\n have h3 : s \u2208 Univ Nat := elt_Univ s\n obtain (m : Nat) (h4 : enum A s m)\n (h5 : \u2200 (y_1 y_2 : Nat), enum A s y_1 \u2192 enum A s y_2 \u2192 y_1 = y_2) from ih h3\n obtain (n : Nat) (h6 : n \u2208 A \u2227 n \u2265 m + 1) from h1 (m + 1)\n obtain (t : Nat) (h7 : num_elts_below A n t) from neb_exists A n\n have h8 : s < t := neb_increase h4 h6.right h7\n have h9 : s + 1 < t \u2228 s + 1 = t := Nat.lt_or_eq_of_le h8\n by_cases on h9\n \u00b7 -- Case 1. h9 : s + 1 < t\n show \u2203 (y : Nat), enum A (s + 1) y from enum_not_skip h7 (s + 1) h9\n done\n \u00b7 -- Case 2. h9 : s + 1 = t\n rewrite [h9]\n apply Exists.intro n\n define\n show n \u2208 A \u2227 num_elts_below A n t from And.intro h6.left h7\n done\n done\n \u00b7 -- Uniqueness\n show \u2200 (y_1 y_2 : Nat), enum A (s + 1) y_1 \u2192 enum A (s + 1) y_2 \u2192 y_1 = y_2\n from enum_unique A (s + 1)\n done\n done\n done\n\nlemma unbdd_subset_nat_match {A : Set Nat}\n (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n matching (enum A) (Univ Nat) A := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix s : Nat; fix n : Nat\n assume h2 : enum A s n\n define at h2\n apply And.intro (elt_Univ s) h2.left\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on (enum A)\n show fcnl_on (enum A) (Univ Nat) from enum_fcnl_of_unbdd h1\n done\n \u00b7 -- Proof of fcnl_on (invRel (enum A))\n show fcnl_on (invRel (enum A)) A from inv_enum_fcnl A\n done\n done\n done\n\nlemma unbdd_subset_nat {A : Set Nat}\n (h1 : \u2200 (m : Nat), \u2203 n \u2208 A, n \u2265 m) :\n denum A := Exists.intro (enum A) (unbdd_subset_nat_match h1)\n\nlemma subset_nat_ctble (A : Set Nat) : ctble A := by\n define --Goal : finite A \u2228 denum A\n by_cases h1 : \u2203 (m : Nat), \u2200 n \u2208 A, n < m\n \u00b7 -- Case 1. h1 : \u2203 (m : Nat), \u2200 n \u2208 A, n < m\n apply Or.inl --Goal : finite A\n obtain (m : Nat) (h2 : \u2200 n \u2208 A, n < m) from h1\n obtain (s : Nat) (h3 : num_elts_below A m s) from neb_exists A m\n apply Exists.intro s\n show I s \u223c A from bdd_subset_nat h2 h3\n done\n \u00b7 -- Case 2. h1 : \u00ac\u2203 (m : Nat), \u2200 n \u2208 A, n < m\n apply Or.inr --Goal : denum A\n push_neg at h1\n --This tactic converts h1 to \u2200 (m : Nat), \u2203 n \u2208 A, m \u2264 n\n show denum A from unbdd_subset_nat h1\n done\n done\n\nlemma ctble_of_equinum_ctble {U V : Type} {A : Set U} {B : Set V}\n (h1 : A \u223c B) (h2 : ctble A) : ctble B := sorry\n\nlemma ctble_iff_equinum_set_nat {U : Type} (A : Set U) :\n ctble A \u2194 \u2203 (I : Set Nat), I \u223c A := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : ctble A\n define at h1 --h1 : finite A \u2228 denum A\n by_cases on h1\n \u00b7 -- Case 1. h1 : finite A\n define at h1 --h1 : \u2203 (n : Nat), I n \u223c A\n obtain (n : Nat) (h2 : I n \u223c A) from h1\n show \u2203 (I : Set Nat), I \u223c A from Exists.intro (I n) h2\n done\n \u00b7 -- Case 2. h1 : denum A\n rewrite [denum_def] at h1 --h1 : Univ Nat \u223c A\n show \u2203 (I : Set Nat), I \u223c A from Exists.intro (Univ Nat) h1\n done\n done\n \u00b7 -- (\u2190)\n assume h1 : \u2203 (I : Set Nat), I \u223c A\n obtain (I : Set Nat) (h2 : I \u223c A) from h1\n have h3 : ctble I := subset_nat_ctble I\n show ctble A from ctble_of_equinum_ctble h2 h3\n done\n done\n\ntheorem Theorem_8_1_5_1_to_2 {U : Type} {A : Set U} (h1 : ctble A) :\n \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A := by\n rewrite [ctble_iff_equinum_set_nat] at h1\n obtain (I : Set Nat) (h2 : I \u223c A) from h1\n obtain (R : Rel Nat U) (h3 : matching R I A) from h2\n define at h3\n --h3 : rel_within R I A \u2227 fcnl_on R I \u2227 fcnl_on (invRel R) A\n apply Exists.intro R\n define --Goal : unique_val_on_N R \u2227 nat_rel_onto R A\n apply And.intro\n \u00b7 -- Proof of unique_val_on_N R\n define\n fix n : Nat; fix x1 : U; fix x2 : U\n assume h4 : R n x1\n assume h5 : R n x2 --Goal : x1 = x2\n have h6 : n \u2208 I \u2227 x1 \u2208 A := h3.left h4\n show x1 = x2 from fcnl_unique h3.right.left h6.left h4 h5\n done\n \u00b7 -- Proof of nat_rel_onto R A\n define\n fix x : U\n assume h4 : x \u2208 A --Goal : \u2203 (n : Nat), R n x\n show \u2203 (n : Nat), R n x from fcnl_exists h3.right.right h4\n done\n done\n\nlemma exists_least_rel_to {U : Type} {S : Rel Nat U} {x : U}\n (h1 : \u2203 (n : Nat), S n x) : \u2203 (n : Nat), least_rel_to S x n := by\n set W : Set Nat := {n : Nat | S n x}\n have h2 : \u2203 (n : Nat), n \u2208 W := h1\n show \u2203 (n : Nat), least_rel_to S x n from well_ord_princ W h2\n done\n\ntheorem Theorem_8_1_5_2_to_3 {U : Type} {A : Set U}\n (h1 : \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A) :\n \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A := by\n obtain (S : Rel Nat U) (h2 : fcnl_onto_from_nat S A) from h1\n define at h2 --h2 : unique_val_on_N S \u2227 nat_rel_onto S A\n set R : Rel U Nat := least_rel_to S\n apply Exists.intro R\n define\n apply And.intro\n \u00b7 -- Proof of fcnl_on R A\n define\n fix x : U\n assume h4 : x \u2208 A --Goal : \u2203! (y : Nat), R x y\n exists_unique\n \u00b7 -- Existence\n have h5 : \u2203 (n : Nat), S n x := h2.right h4\n show \u2203 (n : Nat), R x n from exists_least_rel_to h5\n done\n \u00b7 -- Uniqueness\n fix n1 : Nat; fix n2 : Nat\n assume h5 : R x n1\n assume h6 : R x n2 --Goal : n1 = n2\n define at h5 --h5 : S n1 x \u2227 \u2200 (m : Nat), S m x \u2192 n1 \u2264 m\n define at h6 --h6 : S n2 x \u2227 \u2200 (m : Nat), S m x \u2192 n2 \u2264 m\n have h7 : n1 \u2264 n2 := h5.right n2 h6.left\n have h8 : n2 \u2264 n1 := h6.right n1 h5.left\n linarith\n done\n done\n \u00b7 -- Proof of one-to-one\n fix x1 : U; fix x2 : U; fix n : Nat\n assume h4 : x1 \u2208 A \u2227 R x1 n\n assume h5 : x2 \u2208 A \u2227 R x2 n\n have h6 : R x1 n := h4.right\n have h7 : R x2 n := h5.right\n define at h6 --h6 : S n x1 \u2227 \u2200 (m : Nat), S m x1 \u2192 n \u2264 m\n define at h7 --h7 : S n x2 \u2227 \u2200 (m : Nat), S m x2 \u2192 n \u2264 m\n show x1 = x2 from h2.left h6.left h7.left\n done\n done\n\ntheorem Theorem_8_1_5_3_to_1 {U : Type} {A : Set U}\n (h1 : \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A) :\n ctble A := by\n obtain (S : Rel U Nat) (h2 : fcnl_one_one_to_nat S A) from h1\n define at h2 --h2 : fcnl_on S A \u2227 \u2200 \u2983x1 x2 : U\u2984 \u2983n : Nat\u2984,\n --x1 \u2208 A \u2227 S x1 n \u2192 x2 \u2208 A \u2227 S x2 n \u2192 x1 = x2\n rewrite [ctble_iff_equinum_set_nat] --Goal : \u2203 (I : Set Nat), I \u223c A\n set R : Rel Nat U := invRel (restrict_to S A)\n set I : Set Nat := {n : Nat | \u2203 (x : U), R n x}\n apply Exists.intro I\n define --Goal : \u2203 (R : Rel Nat U), matching R I A\n apply Exists.intro R\n define\n apply And.intro\n \u00b7 -- Proof of rel_within R I A\n define\n fix n : Nat; fix x : U\n assume h3 : R n x --Goal : n \u2208 I \u2227 x \u2208 A\n apply And.intro\n \u00b7 -- Proof that n \u2208 I\n define --Goal : \u2203 (x : U), R n x\n show \u2203 (x : U), R n x from Exists.intro x h3\n done\n \u00b7 -- Proof that x \u2208 A\n define at h3 --h3 : x \u2208 A \u2227 S x n\n show x \u2208 A from h3.left\n done\n done\n \u00b7 -- Proofs of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on R I\n define\n fix n : Nat\n assume h3 : n \u2208 I --Goal : \u2203! (y : U), R n y\n exists_unique\n \u00b7 -- Existence\n define at h3 --h3 : \u2203 (x : U), R n x\n show \u2203 (y : U), R n y from h3\n done\n \u00b7 -- Uniqueness\n fix x1 : U; fix x2 : U\n assume h4 : R n x1\n assume h5 : R n x2\n define at h4 --h4 : x1 \u2208 A \u2227 S x1 n;\n define at h5 --h5 : x2 \u2208 A \u2227 S x2 n\n show x1 = x2 from h2.right h4 h5\n done\n done\n \u00b7 -- Proof of fcnl_on (invRel R) A\n define\n fix x : U\n assume h3 : x \u2208 A --Goal : \u2203! (y : Nat), invRel R x y\n exists_unique\n \u00b7 -- Existence\n obtain (y : Nat) (h4 : S x y) from fcnl_exists h2.left h3\n apply Exists.intro y\n define\n show x \u2208 A \u2227 S x y from And.intro h3 h4\n done\n \u00b7 -- Uniqueness\n fix n1 : Nat; fix n2 : Nat\n assume h4 : invRel R x n1\n assume h5 : invRel R x n2 --Goal : n1 = n2\n define at h4 --h4 : x \u2208 A \u2227 S x n1\n define at h5 --h5 : x \u2208 A \u2227 S x n2\n show n1 = n2 from fcnl_unique h2.left h3 h4.right h5.right\n done\n done\n done\n done\n\ntheorem Theorem_8_1_5_2 {U : Type} (A : Set U) :\n ctble A \u2194 \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : ctble A\n show \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A from\n Theorem_8_1_5_1_to_2 h1\n done\n \u00b7 -- (\u2190)\n assume h1 : \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A\n have h2 : \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A :=\n Theorem_8_1_5_2_to_3 h1\n show ctble A from Theorem_8_1_5_3_to_1 h2\n done\n done\n\ntheorem Theorem_8_1_5_3 {U : Type} (A : Set U) :\n ctble A \u2194 \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : ctble A\n have h2 : \u2203 (R : Rel Nat U), fcnl_onto_from_nat R A :=\n Theorem_8_1_5_1_to_2 h1\n show \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A from\n Theorem_8_1_5_2_to_3 h2\n done\n \u00b7 -- (\u2190)\n assume h1 : \u2203 (R : Rel U Nat), fcnl_one_one_to_nat R A\n show ctble A from Theorem_8_1_5_3_to_1 h1\n done\n done\n\nlemma fqn_def (q : Rat) : fqn q = fnnn (fzn q.num, q.den) := by rfl\n\nlemma fqn_one_one : one_to_one fqn := by\n define\n fix q1 : Rat; fix q2 : Rat\n assume h1 : fqn q1 = fqn q2\n rewrite [fqn_def, fqn_def] at h1\n --h1 : fnnn (fzn q1.num, q1.den) = fnnn (fzn q2.num, q2.den)\n have h2 : (fzn q1.num, q1.den) = (fzn q2.num, q2.den) :=\n fnnn_one_one _ _ h1\n have h3 : fzn q1.num = fzn q2.num \u2227 q1.den = q2.den :=\n Prod.mk.inj h2\n have h4 : q1.num = q2.num := fzn_one_one _ _ h3.left\n show q1 = q2 from Rat.ext h4 h3.right\n done\n\nlemma image_fqn_unbdd :\n \u2200 (m : Nat), \u2203 n \u2208 image fqn (Univ Rat), n \u2265 m := by\n fix m : Nat\n set n : Nat := fqn \u2191m\n apply Exists.intro n\n apply And.intro\n \u00b7 -- Proof that n \u2208 image fqn (Univ Rat)\n define\n apply Exists.intro \u2191m\n apply And.intro (elt_Univ (\u2191m : Rat))\n rfl\n done\n \u00b7 -- Proof that n \u2265 m\n show n \u2265 m from\n calc n\n _ = tri (2 * m + 1) + 2 * m := by rfl\n _ \u2265 m := by linarith\n done\n done\n\ntheorem Theorem_8_1_6 : denum (Univ Rat) := by\n set I : Set Nat := image fqn (Univ Rat)\n have h1 : Univ Nat \u223c I := unbdd_subset_nat image_fqn_unbdd\n have h2 : image fqn (Univ Rat) = I := by rfl\n have h3 : Univ Rat \u223c I :=\n equinum_image (one_one_on_of_one_one fqn_one_one (Univ Rat)) h2\n have h4 : I \u223c Univ Rat := Theorem_8_1_3_2 h3\n show denum (Univ Rat) from Theorem_8_1_3_3 h1 h4\n done\n\n/- Section 8.1\u00bd -/\nlemma eq_zero_of_I_zero_equinum {n : Nat} (h1 : I 0 \u223c I n) : n = 0 := by\n rewrite [\u2190numElts_def, zero_elts_iff_empty] at h1\n --h1 : empty (I n)\n contradict h1 with h2 --Goal : \u2203 (x : Nat), x \u2208 I n\n apply Exists.intro 0\n define\n show 0 < n from Nat.pos_of_ne_zero h2\n done\n\ntheorem eq_of_I_equinum : \u2200 \u2983m n : Nat\u2984, I m \u223c I n \u2192 m = n := by\n by_induc\n \u00b7 -- Base Case\n fix n : Nat\n assume h1 : I 0 \u223c I n\n show 0 = n from (eq_zero_of_I_zero_equinum h1).symm\n done\n \u00b7 -- Induction Step\n fix m : Nat\n assume ih : \u2200 \u2983n : Nat\u2984, I m \u223c I n \u2192 m = n\n fix n : Nat\n assume h1 : I (m + 1) \u223c I n --Goal : m + 1 = n\n have h2 : n \u2260 0 := by\n by_contra h2\n have h3 : I n \u223c I (m + 1) := Theorem_8_1_3_2 h1\n rewrite [h2] at h3\n have h4 : m + 1 = 0 := eq_zero_of_I_zero_equinum h3\n linarith\n done\n obtain (k : Nat) (h3 : n = k + 1) from exists_eq_add_one_of_ne_zero h2\n rewrite [h3] at h1 --h1 : I (m + 1) \u223c I (k + 1)\n rewrite [h3] --Goal : m + 1 = k + 1\n have h4 : m \u2208 I (m + 1) := I_max m\n have h5 : k \u2208 I (k + 1) := I_max k\n have h6 : I (m + 1) \\ {m} \u223c I (k + 1) \\ {k} :=\n remove_one_equinum h1 h4 h5\n rewrite [I_diff, I_diff] at h6 --h6 : I m \u223c I k\n have h7 : m = k := ih h6\n rewrite [h7]\n rfl\n done\n done\n\ntheorem numElts_unique {U : Type} {A : Set U} {m n : Nat}\n (h1 : numElts A m) (h2 : numElts A n) : m = n := by\n rewrite [numElts_def] at h1 --h1 : I m \u223c A\n rewrite [numElts_def] at h2 --h2 : I n \u223c A\n have h3 : A \u223c I n := Theorem_8_1_3_2 h2\n have h4 : I m \u223c I n := Theorem_8_1_3_3 h1 h3\n show m = n from eq_of_I_equinum h4\n done\n\nlemma Set_rp_below_def (a m : Nat) :\n a \u2208 Set_rp_below m \u2194 rel_prime m a \u2227 a < m := by rfl\n\nlemma neb_nrpb (m : Nat) : \u2200 \u2983k : Nat\u2984, k \u2264 m \u2192\n num_elts_below (Set_rp_below m) k (num_rp_below m k) := sorry\n\nlemma neb_phi (m : Nat) :\n num_elts_below (Set_rp_below m) m (phi m) := by\n rewrite [phi_def]\n have h1 : m \u2264 m := by linarith\n show num_elts_below (Set_rp_below m) m (num_rp_below m m) from\n neb_nrpb m h1\n done\n\nlemma phi_is_numElts (m : Nat) :\n numElts (Set_rp_below m) (phi m) := by\n rewrite [numElts_def] --Goal : I (phi m) \u223c Set_rp_below m\n have h1 : \u2200 n \u2208 Set_rp_below m, n < m := by\n fix n : Nat\n assume h2 : n \u2208 Set_rp_below m\n define at h2\n show n < m from h2.right\n done\n have h2 : num_elts_below (Set_rp_below m) m (phi m) := neb_phi m\n show I (phi m) \u223c Set_rp_below m from bdd_subset_nat h1 h2\n done\n\nlemma Lemma_7_4_7_aux {m n : Nat} {s t : Int}\n (h : s * m + t * n = 1) (a b : Nat) :\n t * n * a + s * m * b \u2261 a (MOD m) := by\n define\n apply Exists.intro (s * (b - a))\n show t * n * a + s * m * b - a = m * (s * (b - a)) from\n calc t * n * a + s * m * b - a\n _ = (t * n - 1) * a + s * m * b := by ring\n _ = (t * n - (s * m + t * n)) * a + s * m * b := by rw [h]\n _ = m * (s * (b - a)) := by ring\n done\n\nlemma Lemma_7_4_7 {m n : Nat} [NeZero m] [NeZero n]\n (h1 : rel_prime m n) (a b : Nat) :\n \u2203 (r : Nat), r < m * n \u2227 r \u2261 a (MOD m) \u2227 r \u2261 b (MOD n) := by\n set s : Int := gcd_c1 m n\n set t : Int := gcd_c2 m n\n have h4 : s * m + t * n = gcd m n := gcd_lin_comb n m\n define at h1 --h1 : gcd m n = 1\n rewrite [h1, Nat.cast_one] at h4 --h4 : s * m + t * n = 1\n set x : Int := t * n * a + s * m * b\n have h5 : x \u2261 a (MOD m) := Lemma_7_4_7_aux h4 a b\n rewrite [add_comm] at h4 --h4 : t * n + s * m = 1\n have h6 : s * m * b + t * n * a \u2261 b (MOD n) :=\n Lemma_7_4_7_aux h4 b a\n have h7 : s * m * b + t * n * a = x := by ring\n rewrite [h7] at h6 --h6 : x \u2261 b (MOD n)\n have h8 : m * n \u2260 0 := mul_ne_zero (NeZero.ne m) (NeZero.ne n)\n rewrite [\u2190neZero_iff] at h8 --h8 : NeZero (m * n)\n have h9 : 0 \u2264 x % \u2191(m * n) \u2227 x % \u2191(m * n) < \u2191(m * n) \u2227\n x \u2261 x % \u2191(m * n) (MOD m * n) := mod_cmpl_res (m * n) x\n have h10 : x % \u2191(m * n) < \u2191(m * n) \u2227\n x \u2261 x % \u2191(m * n) (MOD m * n) := h9.right\n set r : Nat := Int.toNat (x % \u2191(m * n))\n have h11 : x % \u2191(m * n) = \u2191r := (Int.toNat_of_nonneg h9.left).symm\n rewrite [h11, Nat.cast_lt] at h10 --h10 : r < m * n \u2227 x \u2261 r (MOD m * n)\n apply Exists.intro r\n apply And.intro h10.left\n have h12 : r \u2261 x (MOD (m * n)) := congr_symm h10.right\n rewrite [Lemma_7_4_5 _ _ h1] at h12 --h12 : r \u2261 x (MOD m) \u2227 r \u2261 x (MOD n)\n apply And.intro\n \u00b7 -- Proof that r \u2261 a (MOD m)\n show r \u2261 a (MOD m) from congr_trans h12.left h5\n done\n \u00b7 -- Proof that r \u2261 b (MOD n)\n show r \u2261 b (MOD n) from congr_trans h12.right h6\n done\n done\n\nlemma Set_prod_def {U V : Type} (A : Set U) (B : Set V) (a : U) (b : V) :\n (a, b) \u2208 A \u00d7\u209b B \u2194 a \u2208 A \u2227 b \u2208 B := by rfl\n\nlemma Rel_prod_def {U V W X : Type} (R : Rel U V) (S : Rel W X)\n (u : U) (v : V) (w : W) (x : X) :\n (R \u00d7\u1d63 S) (u, w) (v, x) \u2194 R u v \u2227 S w x := by rfl\n\nlemma prod_match {U V W X : Type}\n {A : Set U} {B : Set V} {C : Set W} {D : Set X}\n {R : Rel U V} {S : Rel W X}\n (h1 : matching R A B) (h2 : matching S C D) :\n matching (R \u00d7\u1d63 S) (A \u00d7\u209b C) (B \u00d7\u209b D) := sorry\n\ntheorem Theorem_8_1_2_1\n {U V W X : Type} {A : Set U} {B : Set V} {C : Set W} {D : Set X}\n (h1 : A \u223c B) (h2 : C \u223c D) : A \u00d7\u209b C \u223c B \u00d7\u209b D := by\n obtain (R : Rel U V) (h3 : matching R A B) from h1\n obtain (S : Rel W X) (h4 : matching S C D) from h2\n apply Exists.intro (R \u00d7\u1d63 S)\n show matching (R \u00d7\u1d63 S) (A \u00d7\u209b C) (B \u00d7\u209b D) from prod_match h3 h4\n done\n\nlemma qr_def (n a : Nat) : qr n a = (a / n, a % n) := by rfl\n\nlemma qr_one_one (n : Nat) : one_to_one (qr n) := by\n define\n fix a1 : Nat; fix a2 : Nat\n assume h1 : qr n a1 = qr n a2 --Goal : a1 = a2\n rewrite [qr_def, qr_def] at h1\n have h2 : a1 / n = a2 / n \u2227 a1 % n = a2 % n := Prod.mk.inj h1\n show a1 = a2 from\n calc a1\n _ = n * (a1 / n) + a1 % n := (Nat.div_add_mod a1 n).symm\n _ = n * (a2 / n) + a2 % n := by rw [h2.left, h2.right]\n _ = a2 := Nat.div_add_mod a2 n\n done\n\nlemma qr_image (m n : Nat) :\n image (qr n) (I (m * n)) = I m \u00d7\u209b I n := sorry\n\nlemma I_prod (m n : Nat) : I (m * n) \u223c I m \u00d7\u209b I n := equinum_image\n (one_one_on_of_one_one (qr_one_one n) (I (m * n))) (qr_image m n)\n\ntheorem numElts_prod {U V : Type} {A : Set U} {B : Set V} {m n : Nat}\n (h1 : numElts A m) (h2 : numElts B n) : numElts (A \u00d7\u209b B) (m * n) := by\n rewrite [numElts_def] at h1 --h1 : I m \u223c A\n rewrite [numElts_def] at h2 --h2 : I n \u223c B\n rewrite [numElts_def] --Goal : I (m * n) \u223c A \u00d7\u209b B\n have h3 : I m \u00d7\u209b I n \u223c A \u00d7\u209b B := Theorem_8_1_2_1 h1 h2\n have h4 : I (m * n) \u223c I m \u00d7\u209b I n := I_prod m n\n show I (m * n) \u223c A \u00d7\u209b B from Theorem_8_1_3_3 h4 h3\n done\n\nlemma mod_mod_def (m n a : Nat) : mod_mod m n a = (a % m, a % n) := by rfl\n\n--From exercises of Section 7.3\ntheorem congr_rel_prime {m a b : Nat} (h1 : a \u2261 b (MOD m)) :\n rel_prime m a \u2194 rel_prime m b := sorry\n\ntheorem rel_prime_mod (m a : Nat) :\n rel_prime m (a % m) \u2194 rel_prime m a := sorry\n\n/- Stated in Chap7.lean\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n-/\n\n--From exercises of Section 7.4\nlemma Lemma_7_4_6 {a b c : Nat} :\n rel_prime (a * b) c \u2194 rel_prime a c \u2227 rel_prime b c := sorry\n\nlemma left_NeZero_of_mul {m n : Nat} (h : m * n \u2260 0) : NeZero m :=\n neZero_iff.rtl (left_ne_zero_of_mul h)\n\nlemma right_NeZero_of_mul {m n : Nat} (h : m * n \u2260 0) : NeZero n :=\n neZero_iff.rtl (right_ne_zero_of_mul h)\n\nlemma mod_mod_one_one_on {m n : Nat} (h1 : rel_prime m n) :\n one_one_on (mod_mod m n) (Set_rp_below (m * n)) := by\n define\n fix a1 : Nat; fix a2 : Nat\n assume h2 : a1 \u2208 Set_rp_below (m * n)\n assume h3 : a2 \u2208 Set_rp_below (m * n)\n assume h4 : mod_mod m n a1 = mod_mod m n a2 --Goal : a1 = a2\n define at h2; define at h3\n rewrite [mod_mod_def, mod_mod_def] at h4\n have h5 : a1 % m = a2 % m \u2227 a1 % n = a2 % n := Prod.mk.inj h4\n have h6 : m * n \u2260 0 := by linarith\n have h7 : NeZero m := left_NeZero_of_mul h6\n have h8 : NeZero n := right_NeZero_of_mul h6\n rewrite [\u2190congr_iff_mod_eq_Nat, \u2190congr_iff_mod_eq_Nat] at h5\n --h5 : \u2191a1 \u2261 \u2191a2 (MOD m) \u2227 \u2191a1 \u2261 \u2191a2 (MOD n)\n rewrite [\u2190Lemma_7_4_5 _ _ h1] at h5 --h5 : \u2191a1 \u2261 \u2191a2 (MOD m * n)\n rewrite [congr_iff_mod_eq_Nat] at h5 --h5 : a1 % (m * n) = a2 % (m * n)\n rewrite [Nat.mod_eq_of_lt h2.right, Nat.mod_eq_of_lt h3.right] at h5\n show a1 = a2 from h5\n done\n\nlemma mod_elt_Set_rp_below {a m : Nat} [NeZero m] (h1 : rel_prime m a) :\n a % m \u2208 Set_rp_below m := by\n define --Goal : rel_prime m (a % m) \u2227 a % m < m\n rewrite [rel_prime_mod] --Goal : rel_prime m a \u2227 a % m < m\n show rel_prime m a \u2227 a % m < m from\n And.intro h1 (mod_nonzero_lt a (NeZero.ne m))\n done\n\nlemma mod_mod_image {m n : Nat} (h1 : rel_prime m n) :\n image (mod_mod m n) (Set_rp_below (m * n)) =\n (Set_rp_below m) \u00d7\u209b (Set_rp_below n) := by\n apply Set.ext\n fix (b, c) : Nat \u00d7 Nat\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : (b, c) \u2208 image (mod_mod m n) (Set_rp_below (m * n))\n define at h2\n obtain (a : Nat)\n (h3 : a \u2208 Set_rp_below (m * n) \u2227 mod_mod m n a = (b, c)) from h2\n rewrite [Set_rp_below_def, mod_mod_def] at h3\n have h4 : rel_prime (m * n) a := h3.left.left\n rewrite [Lemma_7_4_6] at h4 --h4 : rel_prime m a \u2227 rel_prime n a\n have h5 : a % m = b \u2227 a % n = c := Prod.mk.inj h3.right\n define\n rewrite [\u2190h5.left, \u2190h5.right]\n --Goal : a % m \u2208 Set_rp_below m \u2227 a % n \u2208 Set_rp_below n\n have h6 : m * n \u2260 0 := by linarith\n have h7 : NeZero m := left_NeZero_of_mul h6\n have h8 : NeZero n := right_NeZero_of_mul h6\n apply And.intro\n \u00b7 -- Proof that a % m \u2208 Set_rp_below m\n show a % m \u2208 Set_rp_below m from mod_elt_Set_rp_below h4.left\n done\n \u00b7 -- Proof that a % n \u2208 Set_rp_below n\n show a % n \u2208 Set_rp_below n from mod_elt_Set_rp_below h4.right\n done\n done\n \u00b7 -- (\u2190)\n assume h2 : (b, c) \u2208 Set_rp_below m \u00d7\u209b Set_rp_below n\n rewrite [Set_prod_def, Set_rp_below_def, Set_rp_below_def] at h2\n --h2 : (rel_prime m b \u2227 b < m) \u2227 (rel_prime n c \u2227 c < n)\n define\n have h3 : m \u2260 0 := by linarith\n have h4 : n \u2260 0 := by linarith\n rewrite [\u2190neZero_iff] at h3\n rewrite [\u2190neZero_iff] at h4\n obtain (a : Nat) (h5 : a < m * n \u2227 a \u2261 b (MOD m) \u2227 a \u2261 c (MOD n))\n from Lemma_7_4_7 h1 b c\n apply Exists.intro a\n apply And.intro\n \u00b7 -- Proof of a \u2208 Set_rp_below (m * n)\n define --Goal : rel_prime (m * n) a \u2227 a < m * n\n apply And.intro _ h5.left\n rewrite [Lemma_7_4_6] --Goal : rel_prime m a \u2227 rel_prime n a\n rewrite [congr_rel_prime h5.right.left,\n congr_rel_prime h5.right.right]\n show rel_prime m b \u2227 rel_prime n c from\n And.intro h2.left.left h2.right.left\n done\n \u00b7 -- Proof of mod_mod m n a = (b, c)\n rewrite [congr_iff_mod_eq_Nat, congr_iff_mod_eq_Nat] at h5\n rewrite [mod_mod_def, h5.right.left, h5.right.right]\n --Goal : (b % m, c % n) = (b, c)\n rewrite [Nat.mod_eq_of_lt h2.left.right,\n Nat.mod_eq_of_lt h2.right.right]\n rfl\n done\n done\n done\n\n", "theoremStatement": "lemma Set_rp_below_prod {m n : Nat} (h1 : rel_prime m n) :\n Set_rp_below (m * n) \u223c (Set_rp_below m) \u00d7\u209b (Set_rp_below n) ", "theoremName": "HTPI.Set_rp_below_prod", "fileCreated": {"commit": "fe2721d65e0771872fe4cb8117517e643ac1dc1f", "date": "2023-05-10"}, 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"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", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.Tactic.Simp", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "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.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.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "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.Meta.Tactic.Delta", "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.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "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.Meta.Tactic.Symm", "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.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "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", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6", "HTPILib.Chap7"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n equinum_image (mod_mod_one_one_on h1) (mod_mod_image h1)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 61}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.IntroLean\nnamespace HTPI\n\n/- Definitions -/\ndef even (n : Int) : Prop := \u2203 (k : Int), n = 2 * k\n\ndef odd (n : Int) : Prop := \u2203 (k : Int), n = 2 * k + 1\n\n/- Sections 3.1 and 3.2 -/\ntheorem Example_3_2_4_v2 (P Q R : Prop)\n (h : P \u2192 (Q \u2192 R)) : \u00acR \u2192 (P \u2192 \u00acQ) := by\n assume h2 : \u00acR\n assume h3 : P\n by_contra h4\n have h5 : Q \u2192 R := h h3\n have h6 : R := h5 h4\n show False from h2 h6\n done\n\ntheorem Example_3_2_4_v3 (P Q R : Prop)\n (h : P \u2192 (Q \u2192 R)) : \u00acR \u2192 (P \u2192 \u00acQ) := by\n assume h2 : \u00acR\n assume h3 : P\n by_contra h4\n contradict h2\n show R from h h3 h4\n done\n\ntheorem Like_Example_3_2_5\n (U : Type) (A B C : Set U) (a : U)\n (h1 : a \u2208 A) (h2 : a \u2209 A \\ B)\n (h3 : a \u2208 B \u2192 a \u2208 C) : a \u2208 C := by\n apply h3 _\n define at h2\n demorgan at h2; conditional at h2\n show a \u2208 B from h2 h1\n done\n\n/- Section 3.3 -/\nexample (U : Type) (P Q : Pred U)\n (h1 : \u2200 (x : U), P x \u2192 \u00acQ x)\n (h2 : \u2200 (x : U), Q x) :\n \u00ac\u2203 (x : U), P x := by\n quant_neg --Goal is now \u2200 (x : U), \u00acP x\n fix y : U\n have h3 : P y \u2192 \u00acQ y := h1 y\n have h4 : Q y := h2 y\n contrapos at h3 --Now h3 : Q y \u2192 \u00acP y\n show \u00acP y from h3 h4\n done\n\nexample (U : Type) (A B C : Set U) (h1 : A \u2286 B \u222a C)\n (h2 : \u2200 (x : U), x \u2208 A \u2192 x \u2209 B) : A \u2286 C := by\n define --Goal : \u2200 \u2983a : U\u2984, a \u2208 A \u2192 a \u2208 C\n fix y : U\n assume h3 : y \u2208 A\n have h4 : y \u2209 B := h2 y h3\n define at h1 --h1 : \u2200 \u2983a : U\u2984, a \u2208 A \u2192 a \u2208 B \u222a C\n have h5 : y \u2208 B \u222a C := h1 h3\n define at h5 --h5 : y \u2208 B \u2228 y \u2208 C\n conditional at h5 --h5 : \u00acy \u2208 B \u2192 y \u2208 C\n show y \u2208 C from h5 h4\n done\n\nexample (U : Type) (P Q : Pred U)\n (h1 : \u2200 (x : U), \u2203 (y : U), P x \u2192 \u00ac Q y)\n (h2 : \u2203 (x : U), \u2200 (y : U), P x \u2192 Q y) :\n \u2203 (x : U), \u00acP x := by\n obtain (a : U)\n (h3 : \u2200 (y : U), P a \u2192 Q y) from h2\n have h4 : \u2203 (y : U), P a \u2192 \u00ac Q y := h1 a\n obtain (b : U) (h5 : P a \u2192 \u00ac Q b) from h4\n have h6 : P a \u2192 Q b := h3 b\n apply Exists.intro a _\n by_contra h7\n show False from h5 h7 (h6 h7)\n done\n\ntheorem Example_3_3_5 (U : Type) (B : Set U)\n (F : Set (Set U)) : \u22c3\u2080 F \u2286 B \u2192 F \u2286 \ud835\udcab B := by\n assume h1 : \u22c3\u2080 F \u2286 B\n define\n fix x : Set U\n assume h2 : x \u2208 F\n define\n fix y : U\n assume h3 : y \u2208 x\n define at h1\n apply h1 _\n define\n apply Exists.intro x _\n show x \u2208 F \u2227 y \u2208 x from And.intro h2 h3\n done\n\n/- Section 3.4 -/\ntheorem Like_Example_3_4_1 (U : Type)\n (A B C D : Set U) (h1 : A \u2286 B)\n (h2 : \u00ac\u2203 (c : U), c \u2208 C \u2229 D) :\n A \u2229 C \u2286 B \\ D := by\n define\n fix x : U\n assume h3 : x \u2208 A \u2229 C\n define at h3; define\n apply And.intro\n \u00b7 -- Proof that x \u2208 B.\n show x \u2208 B from h1 h3.left\n done\n \u00b7 -- Proof that x \u2209 D.\n contradict h2 with h4\n apply Exists.intro x\n show x \u2208 C \u2229 D from And.intro h3.right h4\n done\n done\n\nexample (U : Type) (P Q : Pred U)\n (h1 : \u2200 (x : U), P x \u2194 Q x) :\n (\u2203 (x : U), P x) \u2194 \u2203 (x : U), Q x := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : \u2203 (x : U), P x\n obtain (u : U) (h3 : P u) from h2\n have h4 : P u \u2194 Q u := h1 u\n apply Exists.intro u\n show Q u from h4.ltr h3\n done\n \u00b7 -- (\u2190)\n assume h2 : \u2203 (x : U), Q x\n obtain (u : U) (h3 : Q u) from h2\n show \u2203 (x : U), P x from Exists.intro u ((h1 u).rtl h3)\n done\n done\n\ntheorem Example_3_4_5 (U : Type)\n (A B C : Set U) : A \u2229 (B \\ C) = (A \u2229 B) \\ C := by\n apply Set.ext\n fix x : U\n show x \u2208 A \u2229 (B \\ C) \u2194 x \u2208 (A \u2229 B) \\ C from\n calc x \u2208 A \u2229 (B \\ C)\n _ \u2194 x \u2208 A \u2227 (x \u2208 B \u2227 x \u2209 C) := Iff.refl _\n _ \u2194 (x \u2208 A \u2227 x \u2208 B) \u2227 x \u2209 C := and_assoc.symm\n _ \u2194 x \u2208 (A \u2229 B) \\ C := Iff.refl _\n done\n\n/- Section 3.5 -/\ntheorem Example_3_5_2\n (U : Type) (A B C : Set U) :\n A \\ (B \\ C) \u2286 (A \\ B) \u222a C := by\n fix x : U\n assume h1 : x \u2208 A \\ (B \\ C)\n define; define at h1\n have h2 : x \u2209 B \\ C := h1.right\n define at h2; demorgan at h2\n --h2 : x \u2209 B \u2228 x \u2208 C\n by_cases on h2\n \u00b7 -- Case 1. h2 : x \u2209 B\n apply Or.inl\n show x \u2208 A \\ B from And.intro h1.left h2\n done\n \u00b7 -- Case 2. h2 : x \u2208 C\n apply Or.inr\n show x \u2208 C from h2\n done\n done\n\nexample (U : Type) (A B C : Set U)\n (h1 : A \\ B \u2286 C) : A \u2286 B \u222a C := by\n fix x : U\n assume h2 : x \u2208 A\n define\n or_right with h3\n show x \u2208 C from h1 (And.intro h2 h3)\n done\n\nexample\n (U : Type) (A B C : Set U) (h1 : A \u2286 B \u222a C)\n (h2 : \u00ac\u2203 (x : U), x \u2208 A \u2229 B) : A \u2286 C := by\n fix a : U\n assume h3 : a \u2208 A\n quant_neg at h2\n have h4 : a \u2208 B \u222a C := h1 h3\n have h5 : a \u2209 A \u2229 B := h2 a\n define at h4\n define at h5; demorgan at h5\n disj_syll h5 h3 --h5 : \u00aca \u2208 B\n disj_syll h4 h5 --h4 : a \u2208 C\n show a \u2208 C from h4\n done\n\nexample\n (U : Type) (A B C : Set U) (h1 : A \u2286 B \u222a C)\n (h2 : \u00ac\u2203 (x : U), x \u2208 A \u2229 B) : A \u2286 C := by\n fix a : U\n assume h3 : a \u2208 A\n have h4 : a \u2208 B \u222a C := h1 h3\n define at h4\n have h5 : a \u2209 B := by\n contradict h2 with h6\n show \u2203 (x : U), x \u2208 A \u2229 B from\n Exists.intro a (And.intro h3 h6)\n done\n disj_syll h4 h5 --h4 : a \u2208 C\n show a \u2208 C from h4\n done\n\n/- Section 3.6 -/\ntheorem empty_union {U : Type} (B : Set U) :\n \u2205 \u222a B = B := by\n apply Set.ext\n fix x : U\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : x \u2208 \u2205 \u222a B\n define at h1\n have h2 : x \u2209 \u2205 := by\n by_contra h3\n define at h3 --h3 : False\n show False from h3\n done\n disj_syll h1 h2 --h1 : x \u2208 B\n show x \u2208 B from h1\n done\n \u00b7 -- (\u2190)\n assume h1 : x \u2208 B\n show x \u2208 \u2205 \u222a B from Or.inr h1\n done\n done\n\ntheorem union_comm {U : Type} (X Y : Set U) :\n X \u222a Y = Y \u222a X := by\n apply Set.ext\n fix x : U\n define : x \u2208 X \u222a Y\n define : x \u2208 Y \u222a X\n show x \u2208 X \u2228 x \u2208 Y \u2194 x \u2208 Y \u2228 x \u2208 X from or_comm\n done\n\n", "theoremStatement": "theorem Example_3_6_2 (U : Type) :\n \u2203! 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"Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "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.Meta.Tactic.Delta", "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.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "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.Meta.Tactic.Symm", "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.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "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", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro \u2205\n show \u2200 (B : Set U), \u2205 \u222a B = B from empty_union\n done\n \u00b7 -- Uniqueness\n fix C : Set U; fix D : Set U\n assume h1 : \u2200 (B : Set U), C \u222a B = B\n assume h2 : \u2200 (B : Set U), D \u222a B = B\n have h3 : C \u222a D = D := h1 D\n have h4 : D \u222a C = C := h2 C\n show C = D from\n calc C\n _ = D \u222a C := h4.symm\n _ = C \u222a D := union_comm D C\n _ = D := h3\n done\n done", "proofType": "tactic", "proofLengthLines": 18, "proofLengthTokens": 454}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\nlemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done\n\nlemma prod_seq_base {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 0 k f = [1]_m := by rfl\n\nlemma prod_seq_step {m : Nat}\n (n k : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) k f = prod_seq n k f * f (k + n) := by rfl\n\nlemma prod_seq_zero_step {m : Nat}\n (n : Nat) (f : Nat \u2192 ZMod m) :\n prod_seq (n + 1) 0 f = prod_seq n 0 f * f n := by\n rewrite [prod_seq_step, zero_add]\n rfl\n done\n\nlemma prod_one {m : Nat}\n (k : Nat) (f : Nat \u2192 ZMod m) : prod_seq 1 k f = f k := by\n rewrite [prod_seq_step, prod_seq_base, add_zero, mul_comm, Theorem_7_3_6_7]\n rfl\n done\n\nlemma G_def (m a i : Nat) : G m a i = (a * i) % m := by rfl\n\nlemma cc_G (m a i : Nat) : [G m a i]_m = [a]_m * [i]_m :=\n calc [G m a i]_m\n _ = [(a * i) % m]_m := by rfl\n _ = [a * i]_m := (cc_eq_mod m (a * i)).symm\n _ = [a]_m * [i]_m := (mul_class m a i).symm\n\nlemma G_rp_iff {m a : Nat} (h1 : rel_prime m a) (i : Nat) :\n rel_prime m (G m a i) \u2194 rel_prime m i := by\n have h2 : invertible [a]_m := (Theorem_7_3_7 m a).rtl h1\n show rel_prime m (G m a i) \u2194 rel_prime m i from\n calc rel_prime m (G m a i)\n _ \u2194 invertible [G m a i]_m := (Theorem_7_3_7 m (G m a i)).symm\n _ \u2194 invertible ([a]_m * [i]_m) := by rw [cc_G]\n _ \u2194 invertible [i]_m := prod_inv_iff_inv h2 ([i]_m)\n _ \u2194 rel_prime m i := Theorem_7_3_7 m i\n done\n\nlemma FG_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : rel_prime m i) :\n F m (G m a i) = [a]_m * F m i := by\n have h3 : rel_prime m (G m a i) := (G_rp_iff h1 i).rtl h2\n show F m (G m a i) = [a]_m * F m i from\n calc F m (G m a i)\n _ = [G m a i]_m := F_rp_def h3\n _ = [a]_m * [i]_m := cc_G m a i\n _ = [a]_m * F m i := by rw [F_rp_def h2]\n done\n\nlemma FG_not_rp {m a i : Nat} (h1 : rel_prime m a) (h2 : \u00acrel_prime m i) :\n F m (G m a i) = [1]_m := by\n rewrite [\u2190G_rp_iff h1 i] at h2\n show F m (G m a i) = [1]_m from F_not_rp_def h2\n done\n\nlemma FG_prod {m a : Nat} (h1 : rel_prime m a) :\n \u2200 (k : Nat), prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) := by\n by_induc\n \u00b7 -- Base Case\n show prod_seq 0 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) from\n calc prod_seq 0 0 ((F m) \u2218 (G m a))\n _ = [1]_m := prod_seq_base _ _\n _ = [a]_m ^ 0 * [1]_m := by ring\n _ = [a]_m ^ (num_rp_below m 0) * prod_seq 0 0 (F m) := by\n rw [num_rp_below_base, prod_seq_base]\n done\n \u00b7 -- Induction Step\n fix k : Nat\n assume ih : prod_seq k 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m)\n by_cases h2 : rel_prime m k\n \u00b7 -- Case 1. h2 : rel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([a]_m * F m k) := by rw [FG_rp h1 h2]\n _ = [a]_m ^ ((num_rp_below m k) + 1) *\n ((prod_seq k 0 (F m)) * F m k) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_rp h2, prod_seq_zero_step]\n done\n \u00b7 -- Case 2. h2 : \u00acrel_prime m k\n show prod_seq (k + 1) 0 ((F m) \u2218 (G m a)) =\n [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) from\n calc prod_seq (k + 1) 0 ((F m) \u2218 (G m a))\n _ = prod_seq k 0 ((F m) \u2218 (G m a)) *\n F m (G m a k) := prod_seq_zero_step _ _\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n F m (G m a k) := by rw [ih]\n _ = [a]_m ^ (num_rp_below m k) * prod_seq k 0 (F m) *\n ([1]_m) := by rw [FG_not_rp h1 h2]\n _ = [a]_m ^ (num_rp_below m k) *\n (prod_seq k 0 (F m) * ([1]_m)) := by ring\n _ = [a]_m ^ (num_rp_below m (k + 1)) *\n prod_seq (k + 1) 0 (F m) := by\n rw [num_rp_below_step_not_rp h2, prod_seq_zero_step,\n F_not_rp_def h2]\n done\n done\n done\n\nlemma G_maps_below (m a : Nat) [NeZero m] : maps_below m (G m a) := by\n define --Goal : \u2200 i < m, G m a i < m\n fix i : Nat\n assume h1 : i < m\n rewrite [G_def] --Goal : a * i % m < m\n show a * i % m < m from mod_nonzero_lt (a * i) (NeZero.ne m)\n done\n\nlemma left_inv_one_one_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g' (g i) = i) : one_one_below n g := sorry\n\nlemma right_inv_onto_below {n : Nat} {g g' : Nat \u2192 Nat}\n (h1 : \u2200 i < n, g (g' i) = i) (h2 : maps_below n g') :\n onto_below n g := by\n define at h2; define\n fix k : Nat\n assume h3 : k < n\n apply Exists.intro (g' k)\n show g' k < n \u2227 g (g' k) = k from And.intro (h2 k h3) (h1 k h3)\n done\n\nlemma cc_mul_inv_mod_eq_one {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n [a]_m * [inv_mod m a]_m = [1]_m := by\n have h2 : 0 \u2264 (gcd_c2 m a) % m := mod_nonneg m (gcd_c2 m a)\n show [a]_m * [inv_mod m a]_m = [1]_m from\n calc [a]_m * [inv_mod m a]_m\n _ = [a]_m * [Int.toNat ((gcd_c2 m a) % m)]_m := by rfl\n _ = [a]_m * [(gcd_c2 m a) % m]_m := by rw [Int.toNat_of_nonneg h2]\n _ = [a]_m * [gcd_c2 m a]_m := by rw [\u2190cc_eq_mod]\n _ = [1]_m := gcd_c2_inv h1\n done\n\nlemma mul_mod_mod_eq_mul_mod (m a b : Nat) : (a * (b % m)) % m = (a * b) % m :=\n calc a * (b % m) % m\n = a % m * (b % m % m) % m := Nat.mul_mod _ _ _\n _ = a % m * (b % m) % m := by rw [Nat.mod_mod]\n _ = a * b % m := (Nat.mul_mod _ _ _).symm\n\nlemma mod_mul_mod_eq_mul_mod (m a b : Nat) : (a % m * b) % m = (a * b) % m := by\n rewrite [mul_comm, mul_mod_mod_eq_mul_mod, mul_comm]\n rfl\n done\n\ntheorem congr_iff_mod_eq_Nat (m a b : Nat) [NeZero m] :\n \u2191a \u2261 \u2191b (MOD m) \u2194 a % m = b % m := sorry\n\nlemma mul_inv_mod_cancel {m a i : Nat} [NeZero m]\n (h1 : rel_prime m a) (h2 : i < m) : a * (inv_mod m a) * i % m = i := by\n have h3 : [a]_m * [inv_mod m a]_m = [1]_m := cc_mul_inv_mod_eq_one h1\n rewrite [mul_class, cc_eq_iff_congr, \u2190Nat.cast_mul, \u2190Nat.cast_one, congr_iff_mod_eq_Nat] at h3\n show a * inv_mod m a * i % m = i from\n calc a * (inv_mod m a) * i % m\n _ = (a * inv_mod m a) % m * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = 1 % m * i % m := by rw [h3]\n _ = 1 * i % m := by rw [mod_mul_mod_eq_mul_mod]\n _ = i % m := by rw [one_mul]\n _ = i := Nat.mod_eq_of_lt h2\n done\n\nlemma Ginv_def {m a i : Nat} : Ginv m a i = G m (inv_mod m a) i := by rfl\n\nlemma Ginv_right_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, G m a (Ginv m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show G m a (Ginv m a i) = i from\n calc G m a (Ginv m a i)\n _ = a * ((inv_mod m a * i) % m) % m := by rfl\n _ = a * (inv_mod m a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_left_inv {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n \u2200 i < m, Ginv m a (G m a i) = i := by\n fix i : Nat\n assume h2 : i < m\n show Ginv m a (G m a i) = i from\n calc Ginv m a (G m a i)\n _ = inv_mod m a * ((a * i) % m) % m := by rfl\n _ = inv_mod m a * (a * i) % m := by rw [mul_mod_mod_eq_mul_mod]\n _ = a * inv_mod m a * i % m := by rw [\u2190mul_assoc, mul_comm (inv_mod m a)]\n _ = i := mul_inv_mod_cancel h1 h2\n done\n\nlemma Ginv_maps_below (m a : Nat) [NeZero m] :\n maps_below m (Ginv m a) := G_maps_below m (inv_mod m a)\n\nlemma G_one_one_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n one_one_below m (G m a) :=\n left_inv_one_one_below (Ginv_left_inv h1)\n\n", "theoremStatement": "lemma G_onto_below {m a : Nat} [NeZero m] (h1 : rel_prime m a) :\n onto_below m (G m a) ", "theoremName": "HTPI.G_onto_below", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 1355, "tokenPositionInFile": 44041, "theoremPositionInFile": 131}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 7, 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"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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n right_inv_onto_below (Ginv_right_inv h1) (Ginv_maps_below m a)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 67}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap4\nnamespace HTPI\n\n/- Definitions -/\ndef graph {A B : Type} (f : A \u2192 B) : Set (A \u00d7 B) :=\n {(a, b) : A \u00d7 B | f a = b}\n\ndef is_func_graph {A B : Type} (G : Set (A \u00d7 B)) : Prop :=\n \u2200 (x : A), \u2203! (y : B), (x, y) \u2208 G\n\ndef onto {A B : Type} (f : A \u2192 B) : Prop :=\n \u2200 (y : B), \u2203 (x : A), f x = y\n\ndef one_to_one {A B : Type} (f : A \u2192 B) : Prop :=\n \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n\ndef closed {A : Type} (f : A \u2192 A) (C : Set A) : Prop := \u2200 x \u2208 C, f x \u2208 C\n\ndef closure {A : Type} (f : A \u2192 A) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed f D}\n\ndef closed2 {A : Type} (f : A \u2192 A \u2192 A) (C : Set A) : Prop :=\n \u2200 x \u2208 C, \u2200 y \u2208 C, f x y \u2208 C\n\ndef closure2 {A : Type} (f : A \u2192 A \u2192 A) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed2 f D}\n\ndef closed_family {A : Type} (F : Set (A \u2192 A)) (C : Set A) : Prop :=\n \u2200 f \u2208 F, closed f C\n\ndef closure_family {A : Type} (F : Set (A \u2192 A)) (B C : Set A) : Prop :=\n smallestElt (sub A) C {D : Set A | B \u2286 D \u2227 closed_family F D}\n\ndef image {A B : Type} (f : A \u2192 B) (X : Set A) : Set B :=\n {f x | x \u2208 X}\n\ndef inverse_image {A B : Type} (f : A \u2192 B) (Y : Set B) : Set A :=\n {a : A | f a \u2208 Y}\n\n/- Section 5.1 -/\ntheorem graph_def {A B : Type} (f : A \u2192 B) (a : A) (b : B) :\n (a, b) \u2208 graph f \u2194 f a = b := by rfl\n\ntheorem func_from_graph_ltr {A B : Type} (F : Set (A \u00d7 B)) :\n (\u2203 (f : A \u2192 B), graph f = F) \u2192 is_func_graph F := sorry\n\n--This proof is explained in Section 8.2\ntheorem func_from_graph_rtl {A B : Type} (F : Set (A \u00d7 B)) :\n is_func_graph F \u2192 (\u2203 (f : A \u2192 B), graph f = F) := by\n assume h1 : is_func_graph F\n define at h1 --h1 : \u2200 (x : A), \u2203! (y : B), (x, y) \u2208 F\n have h2 : \u2200 (x : A), \u2203 (y : B), (x, y) \u2208 F := by\n fix x : A\n obtain (y : B) (h3 : (x, y) \u2208 F)\n (h4 : \u2200 (y1 y2 : B), (x, y1) \u2208 F \u2192 (x, y2) \u2208 F \u2192 y1 = y2) from h1 x\n show \u2203 (y : B), (x, y) \u2208 F from Exists.intro y h3\n done\n set f : A \u2192 B := fun (x : A) => Classical.choose (h2 x)\n apply Exists.intro f\n apply Set.ext\n fix (x, y) : A \u00d7 B\n have h3 : (x, f x) \u2208 F := Classical.choose_spec (h2 x)\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h4 : (x, y) \u2208 graph f\n define at h4 --h4 : f x = y\n rewrite [h4] at h3\n show (x, y) \u2208 F from h3\n done\n \u00b7 -- (\u2190)\n assume h4 : (x, y) \u2208 F\n define --Goal : f x = y\n obtain (z : B) (h5 : (x, z) \u2208 F)\n (h6 : \u2200 (y1 y2 : B), (x, y1) \u2208 F \u2192 (x, y2) \u2208 F \u2192 y1 = y2) from h1 x\n show f x = y from h6 (f x) y h3 h4\n done\n done\n\ntheorem func_from_graph {A B : Type} (F : Set (A \u00d7 B)) :\n (\u2203 (f : A \u2192 B), graph f = F) \u2194 is_func_graph F :=\n Iff.intro (func_from_graph_ltr F) (func_from_graph_rtl F)\n\ntheorem Theorem_5_1_4 {A B : Type} (f g : A \u2192 B) :\n (\u2200 (a : A), f a = g a) \u2192 f = g := funext\n\nexample {A B : Type} (f g : A \u2192 B) :\n graph f = graph g \u2192 f = g := by\n assume h1 : graph f = graph g --Goal : f = g\n apply funext --Goal : \u2200 (x : A), f x = g x\n fix x : A\n have h2 : (x, f x) \u2208 graph f := by\n define --Goal : f x = f x\n rfl\n done\n rewrite [h1] at h2 --h2 : (x, f x) \u2208 graph g\n define at h2 --h2 : g x = f x\n show f x = g x from h2.symm\n done\n\ndef square1 (n : Nat) : Nat := n ^ 2\n\ndef square2 : Nat \u2192 Nat := fun (n : Nat) => n ^ 2\n\nexample : square1 = square2 := by rfl\n\n#eval square1 7 --Answer: 49\n\ntheorem Theorem_5_1_5 {A B C : Type} (f : A \u2192 B) (g : B \u2192 C) :\n \u2203 (h : A \u2192 C), graph h = comp (graph g) (graph f) := by\n set h : A \u2192 C := fun (x : A) => g (f x)\n apply Exists.intro h\n apply Set.ext\n fix (a, c) : A \u00d7 C\n apply Iff.intro\n \u00b7 -- Proof that (a, c) \u2208 graph h \u2192 (a, c) \u2208 comp (graph g) (graph f)\n assume h1 : (a, c) \u2208 graph h\n define at h1 --h1 : h a = c\n define --Goal : \u2203 (x : B), (a, x) \u2208 graph f \u2227 (x, c) \u2208 graph g\n apply Exists.intro (f a)\n apply And.intro\n \u00b7 -- Proof that (a, f a) \u2208 graph f\n define\n rfl\n done\n \u00b7 -- Proof that (f a, c) \u2208 graph g\n define\n show g (f a) = c from h1\n done\n done\n \u00b7 -- Proof that (a, c) \u2208 comp (graph g) (graph f) \u2192 (a, c) \u2208 graph h\n assume h1 : (a, c) \u2208 comp (graph g) (graph f)\n define --Goal : h a = c\n define at h1 --h1 : \u2203 (x : B), (a, x) \u2208 graph f \u2227 (x, c) \u2208 graph g\n obtain (b : B) (h2 : (a, b) \u2208 graph f \u2227 (b, c) \u2208 graph g) from h1\n have h3 : (a, b) \u2208 graph f := h2.left\n have h4 : (b, c) \u2208 graph g := h2.right\n define at h3 --h3 : f a = b\n define at h4 --h4 : g b = c\n rewrite [\u2190h3] at h4 --h4 : g (f a) = c\n show h a = c from h4\n done\n done\n\nexample {A B C D : Type} (f : A \u2192 B) (g : B \u2192 C) (h : C \u2192 D) :\n h \u2218 (g \u2218 f) = (h \u2218 g) \u2218 f := by rfl\n\nexample {A B : Type} (f : A \u2192 B) : f \u2218 id = f := by rfl\n\nexample {A B : Type} (f : A \u2192 B) : id \u2218 f = f := by rfl\n\n/- Section 5.2 -/\ntheorem Theorem_5_2_5_1 {A B C : Type} (f : A \u2192 B) (g : B \u2192 C) :\n one_to_one f \u2192 one_to_one g \u2192 one_to_one (g \u2218 f) := by\n assume h1 : one_to_one f\n assume h2 : one_to_one g\n define at h1 --h1 : \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n define at h2 --h2 : \u2200 (x1 x2 : B), g x1 = g x2 \u2192 x1 = x2\n define --Goal : \u2200 (x1 x2 : A), (g \u2218 f) x1 = (g \u2218 f) x2 \u2192 x1 = x2\n fix a1 : A\n fix a2 : A --Goal : (g \u2218 f) a1 = (g \u2218 f) a2 \u2192 a1 = a2\n define : (g \u2218 f) a1; define : (g \u2218 f) a2\n --Goal : g (f a1) = g (f a2) \u2192 a1 = a2\n assume h3 : g (f a1) = g (f a2)\n have h4 : f a1 = f a2 := h2 (f a1) (f a2) h3\n show a1 = a2 from h1 a1 a2 h4\n done\n\nlemma comp_def {A B C : Type} (g : B \u2192 C) (f : A \u2192 B) (x : A) :\n (g \u2218 f) x = g (f x) := by rfl\n\ntheorem Theorem_5_2_5_2 {A B C : Type} (f : A \u2192 B) (g : B \u2192 C) :\n onto f \u2192 onto g \u2192 onto (g \u2218 f) := by\n assume h1 : onto f\n assume h2 : onto g\n define at h1 --h1 : \u2200 (y : B), \u2203 (x : A), f x = y\n define at h2 --h2 : \u2200 (y : C), \u2203 (x : B), g x = y\n define --Goal : \u2200 (y : C), \u2203 (x : A), (g \u2218 f) x = y\n fix c : C\n obtain (b : B) (h3 : g b = c) from h2 c\n obtain (a : A) (h4 : f a = b) from h1 b\n apply Exists.intro a --Goal : (g \u2218 f) a = c\n rewrite [comp_def] --Goal : g (f a) = c\n rewrite [\u2190h4] at h3\n show g (f a) = c from h3\n done\n\n/- Section 5.3 -/\ntheorem Theorem_5_3_1 {A B : Type}\n (f : A \u2192 B) (h1 : one_to_one f) (h2 : onto f) :\n \u2203 (g : B \u2192 A), graph g = inv (graph f) := by\n rewrite [func_from_graph] --Goal : is_func_graph (inv (graph f))\n define --Goal : \u2200 (x : B), \u2203! (y : A), (x, y) \u2208 inv (graph f)\n fix b : B\n exists_unique\n \u00b7 -- Existence\n define at h2 --h2 : \u2200 (y : B), \u2203 (x : A), f x = y\n obtain (a : A) (h4 : f a = b) from h2 b\n apply Exists.intro a --Goal : (b, a) \u2208 inv (graph f)\n define --Goal : f a = b\n show f a = b from h4\n done\n \u00b7 -- Uniqueness\n fix a1 : A; fix a2 : A\n assume h3 : (b, a1) \u2208 inv (graph f)\n assume h4 : (b, a2) \u2208 inv (graph f) --Goal : a1 = a2\n define at h3 --h3 : f a1 = b\n define at h4 --h4 : f a2 = b\n rewrite [\u2190h4] at h3 --h3 : f a1 = f a2\n define at h1 --h1 : \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n show a1 = a2 from h1 a1 a2 h3\n done\n done\n\ntheorem Theorem_5_3_2_1 {A B : Type} (f : A \u2192 B) (g : B \u2192 A)\n (h1 : graph g = inv (graph f)) : g \u2218 f = id := by\n apply funext --Goal : \u2200 (x : A), (g \u2218 f) x = id x\n fix a : A --Goal : (g \u2218 f) a = id a\n have h2 : (f a, a) \u2208 graph g := by\n rewrite [h1] --Goal : (f a, a) \u2208 inv (graph f)\n define --Goal : f a = f a\n rfl\n done\n define at h2 --h2 : g (f a) = a\n show (g \u2218 f) a = id a from h2\n done\n\ntheorem Theorem_5_3_2_2 {A B : Type} (f : A \u2192 B) (g : B \u2192 A)\n (h1 : graph g = inv (graph f)) : f \u2218 g = id := sorry\n\n", "theoremStatement": "theorem Theorem_5_3_3_1 {A B : Type} (f : A \u2192 B) (g : B \u2192 A)\n (h1 : g \u2218 f = id) : one_to_one f ", "theoremName": "HTPI.Theorem_5_3_3_1", "fileCreated": {"commit": "cc48e6ba4c656bf74f7690323b2c27f2c466665b", "date": "2023-03-11"}, "theoremCreated": {"commit": "190e34021c4943fc63cee05edf567875671378bb", "date": "2023-02-18"}, "file": "htpi/HTPILib/Chap5.lean", "module": "HTPILib.Chap5", "jsonFile": "HTPILib.Chap5.jsonl", "positionMetadata": {"lineInFile": 232, "tokenPositionInFile": 7813, "theoremPositionInFile": 26}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 13, "importedModules": 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"Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n define --Goal : \u2200 (x1 x2 : A), f x1 = f x2 \u2192 x1 = x2\n fix a1 : A; fix a2 : A\n assume h2 : f a1 = f a2\n show a1 = a2 from\n calc a1\n _ = id a1 := by rfl\n _ = (g \u2218 f) a1 := by rw [h1]\n _ = g (f a1) := by rfl\n _ = g (f a2) := by rw [h2]\n _ = (g \u2218 f) a2 := by rfl\n _ = id a2 := by rw [h1]\n _ = a2 := by rfl\n done", "proofType": "tactic", "proofLengthLines": 13, "proofLengthTokens": 370}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\n", "theoremStatement": "theorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b ", "theoremName": "HTPI.gcd_dvd", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", "positionMetadata": {"lineInFile": 157, "tokenPositionInFile": 4317, "theoremPositionInFile": 33}, 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"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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done", "proofType": "tactic", "proofLengthLines": 22, "proofLengthTokens": 670}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap5\nnamespace HTPI\n\n/- Definitions -/\ndef invRel {U V : Type} (R : Rel U V) : Rel V U :=\n RelFromExt (inv (extension R))\n\ndef rel_within {U V : Type} (R : Rel U V) (A : Set U) (B : Set V) : Prop :=\n \u2200 \u2983x : U\u2984 \u2983y : V\u2984, R x y \u2192 x \u2208 A \u2227 y \u2208 B\n\ndef fcnl_on {U V : Type} (R : Rel U V) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203! (y : V), R x y\n\ndef matching {U V : Type} (R : Rel U V) (A : Set U) (B : Set V) : Prop :=\n rel_within R A B \u2227 fcnl_on R A \u2227 fcnl_on (invRel R) B\n\ndef equinum {U V : Type} (A : Set U) (B : Set V) : Prop :=\n \u2203 (R : Rel U V), matching R A B\n\nnotation:50 A:50 \" \u223c \" B:50 => equinum A B\n\ndef RelWithinFromFunc {U V : Type} (f : U \u2192 V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 f x = y\n\ndef one_one_on {U V : Type} (f : U \u2192 V) (A : Set U) : Prop :=\n \u2200 \u2983x1 x2 : U\u2984, x1 \u2208 A \u2192 x2 \u2208 A \u2192 f x1 = f x2 \u2192 x1 = x2\n\ndef Univ (U : Type) : Set U := {x : U | True}\n\ndef compRel {U V W : Type} (S : Rel V W) (R : Rel U V) : Rel U W :=\n RelFromExt (comp (extension S) (extension R))\n\ndef I (n : Nat) : Set Nat := {k : Nat | k < n}\n\ndef finite {U : Type} (A : Set U) : Prop :=\n \u2203 (n : Nat), I n \u223c A\n\ndef denum {U : Type} (A : Set U) : Prop :=\n Univ Nat \u223c A\n\ndef ctble {U : Type} (A : Set U) : Prop :=\n finite A \u2228 denum A\n\ndef numElts {U : Type} (A : Set U) (n : Nat) : Prop := I n \u223c A\n\ndef emptyRel (U V : Type) (x : U) (y : V) : Prop := False\n\ndef remove_one {U V : Type} (R : Rel U V) (u : U) (v : V)\n (x : U) (y : V) : Prop := x \u2260 u \u2227 y \u2260 v \u2227 (R x y \u2228 (R x v \u2227 R u y))\n\ndef one_match {U V : Type} (a : U) (b : V)\n (x : U) (y : V) : Prop := x = a \u2227 y = b\n\n/- Section 8.1 -/\nlemma invRel_def {U V : Type} (R : Rel U V) (u : U) (v : V) :\n invRel R v u \u2194 R u v := by rfl\n\ntheorem equinum_image {U V : Type} {A : Set U} {B : Set V} {f : U \u2192 V}\n (h1 : one_one_on f A) (h2 : image f A = B) : A \u223c B := by\n rewrite [\u2190h2]\n define --Goal : \u2203 (R : Rel U V), matching R A (image f A)\n set R : Rel U V := RelWithinFromFunc f A\n apply Exists.intro R\n define --Goal : rel_within R A (image f A) \u2227\n --fcnl_on R A \u2227 fcnl_on (invRel R) (image f A)\n apply And.intro\n \u00b7 -- Proof of rel_within\n define --Goal : \u2200 \u2983x : U\u2984 \u2983y : V\u2984, R x y \u2192 x \u2208 A \u2227 y \u2208 image f A\n fix x : U; fix y : V\n assume h3 : R x y --Goal : x \u2208 A \u2227 y \u2208 image f A\n define at h3 --h3 : x \u2208 A \u2227 f x = y\n apply And.intro h3.left\n define\n show \u2203 x \u2208 A, f x = y from Exists.intro x h3\n done\n \u00b7 -- Proofs of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on R A\n define --Goal : \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203! (y : V), R x y\n fix x : U\n assume h3 : x \u2208 A\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (f x)\n define --Goal : x \u2208 A \u2227 f x = f x\n apply And.intro h3\n rfl\n done\n \u00b7 -- Uniqueness\n fix y1 : V; fix y2 : V\n assume h4 : R x y1\n assume h5 : R x y2 --Goal : y1 = y2\n define at h4; define at h5\n --h4 : x \u2208 A \u2227 f x = y1; h5 : x \u2208 A \u2227 f x = y2\n rewrite [h4.right] at h5\n show y1 = y2 from h5.right\n done\n done\n \u00b7 -- Proof of fcnl_on (invRel R) (image f A)\n define --Goal : \u2200 \u2983x : V\u2984, x \u2208 image f A \u2192 \u2203! (y : U), invRel R x y\n fix y : V\n assume h3 : y \u2208 image f A\n obtain (x : U) (h4 : x \u2208 A \u2227 f x = y) from h3\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro x\n define\n show x \u2208 A \u2227 f x = y from h4\n done\n \u00b7 -- Uniqueness\n fix x1 : U; fix x2 : U\n assume h5 : invRel R y x1\n assume h6 : invRel R y x2\n define at h5; define at h6\n --h5 : x1 \u2208 A \u2227 f x1 = y; h6 : x2 \u2208 A \u2227 f x2 = y\n rewrite [\u2190h6.right] at h5\n show x1 = x2 from h1 h5.left h6.left h5.right\n done\n done\n done\n done\n\nlemma id_one_one_on {U : Type} (A : Set U) : one_one_on id A := by\n define\n fix x1 : U; fix x2 : U\n assume h1 : x1 \u2208 A\n assume h2 : x2 \u2208 A\n assume h3 : id x1 = id x2\n show x1 = x2 from h3\n done\n\nlemma image_id {U : Type} (A : Set U) : image id A = A := by\n apply Set.ext\n fix x : U\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : x \u2208 image id A\n obtain (y : U) (h2 : y \u2208 A \u2227 id y = x) from h1\n rewrite [\u2190h2.right]\n show id y \u2208 A from h2.left\n done\n \u00b7 -- (\u2190)\n assume h1 : x \u2208 A\n apply Exists.intro x --Goal : x \u2208 A \u2227 id x = x\n apply And.intro h1\n rfl\n done\n done\n\ntheorem Theorem_8_1_3_1 {U : Type} (A : Set U) : A \u223c A :=\n equinum_image (id_one_one_on A) (image_id A)\n\nlemma inv_inv {U V : Type} (R : Rel U V) : invRel (invRel R) = R := by rfl\n\nlemma inv_match {U V : Type} {R : Rel U V} {A : Set U} {B : Set V}\n (h : matching R A B) : matching (invRel R) B A := by\n define --Goal : rel_within (invRel R) B A \u2227\n --fcnl_on (invRel R) B \u2227 fcnl_on (invRel (invRel R)) A\n define at h --h : rel_within R A B \u2227 fcnl_on R A \u2227 fcnl_on (invRel R) B\n apply And.intro\n \u00b7 -- Proof that rel_within (invRel R) B A\n define --Goal : \u2200 \u2983x : V\u2984 \u2983y : U\u2984, invRel R x y \u2192 x \u2208 B \u2227 y \u2208 A\n fix y : V; fix x : U\n assume h1 : invRel R y x\n define at h1 --h1 : R x y\n have h2 : x \u2208 A \u2227 y \u2208 B := h.left h1\n show y \u2208 B \u2227 x \u2208 A from And.intro h2.right h2.left\n done\n \u00b7 -- proof that fcnl_on (inv R) B \u2227 fcnl_on (inv (inv R)) A\n rewrite [inv_inv]\n show fcnl_on (invRel R) B \u2227 fcnl_on R A from\n And.intro h.right.right h.right.left\n done\n done\n\ntheorem Theorem_8_1_3_2 {U V : Type} {A : Set U} {B : Set V}\n (h : A \u223c B) : B \u223c A := by\n obtain (R : Rel U V) (h1 : matching R A B) from h\n apply Exists.intro (invRel R)\n show matching (invRel R) B A from inv_match h1\n done\n\nlemma fcnl_exists {U V : Type} {R : Rel U V} {A : Set U} {x : U}\n (h1 : fcnl_on R A) (h2 : x \u2208 A) : \u2203 (y : V), R x y := by\n define at h1\n obtain (y : V) (h3 : R x y)\n (h4 : \u2200 (y_1 y_2 : V), R x y_1 \u2192 R x y_2 \u2192 y_1 = y_2) from h1 h2\n show \u2203 (y : V), R x y from Exists.intro y h3\n done\n\nlemma fcnl_unique {U V : Type}\n {R : Rel U V} {A : Set U} {x : U} {y1 y2 : V} (h1 : fcnl_on R A)\n (h2 : x \u2208 A) (h3 : R x y1) (h4 : R x y2) : y1 = y2 := by\n define at h1\n obtain (z : V) (h5 : R x z)\n (h6 : \u2200 (y_1 y_2 : V), R x y_1 \u2192 R x y_2 \u2192 y_1 = y_2) from h1 h2\n show y1 = y2 from h6 y1 y2 h3 h4\n done\n\nlemma compRel_def {U V W : Type}\n (S : Rel V W) (R : Rel U V) (u : U) (w : W) :\n compRel S R u w \u2194 \u2203 (x : V), R u x \u2227 S x w := by rfl\n\nlemma inv_comp {U V W : Type} (R : Rel U V) (S : Rel V W) :\n invRel (compRel S R) = compRel (invRel R) (invRel S) :=\n calc invRel (compRel S R)\n _ = RelFromExt (inv (comp (extension S) (extension R))) := by rfl\n _ = RelFromExt (comp (inv (extension R)) (inv (extension S))) := by\n rw [Theorem_4_2_5_5]\n _ = compRel (invRel R) (invRel S) := by rfl\n\nlemma comp_fcnl {U V W : Type} {R : Rel U V} {S : Rel V W}\n {A : Set U} {B : Set V} {C : Set W} (h1 : matching R A B)\n (h2 : matching S B C) : fcnl_on (compRel S R) A := by\n define; define at h1; define at h2\n fix a : U\n assume h3 : a \u2208 A\n obtain (b : V) (h4 : R a b) from fcnl_exists h1.right.left h3\n have h5 : a \u2208 A \u2227 b \u2208 B := h1.left h4\n obtain (c : W) (h6 : S b c) from fcnl_exists h2.right.left h5.right\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro c\n rewrite [compRel_def]\n show \u2203 (x : V), R a x \u2227 S x c from Exists.intro b (And.intro h4 h6)\n done\n \u00b7 -- Uniqueness\n fix c1 : W; fix c2 : W\n assume h7 : compRel S R a c1\n assume h8 : compRel S R a c2 --Goal : c1 = c2\n rewrite [compRel_def] at h7\n rewrite [compRel_def] at h8\n obtain (b1 : V) (h9 : R a b1 \u2227 S b1 c1) from h7\n obtain (b2 : V) (h10 : R a b2 \u2227 S b2 c2) from h8\n have h11 : b1 = b := fcnl_unique h1.right.left h3 h9.left h4\n have h12 : b2 = b := fcnl_unique h1.right.left h3 h10.left h4\n rewrite [h11] at h9\n rewrite [h12] at h10\n show c1 = c2 from\n fcnl_unique h2.right.left h5.right h9.right h10.right\n done\n done\n\nlemma comp_match {U V W : Type} {R : Rel U V} {S : Rel V W}\n {A : Set U} {B : Set V} {C : Set W} (h1 : matching R A B)\n (h2 : matching S B C) : matching (compRel S R) A C := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within (compRel S R) A C\n define\n fix a : U; fix c : W\n assume h3 : compRel S R a c\n rewrite [compRel_def] at h3\n obtain (b : V) (h4 : R a b \u2227 S b c) from h3\n have h5 : a \u2208 A \u2227 b \u2208 B := h1.left h4.left\n have h6 : b \u2208 B \u2227 c \u2208 C := h2.left h4.right\n show a \u2208 A \u2227 c \u2208 C from And.intro h5.left h6.right\n done\n \u00b7 -- Proof of fcnl_on statements\n apply And.intro\n \u00b7 -- Proof of fcnl_on (compRel S R) A\n show fcnl_on (compRel S R) A from comp_fcnl h1 h2\n done\n \u00b7 -- Proof of fcnl_on (invRel (compRel S R)) Z\n rewrite [inv_comp]\n have h3 : matching (invRel R) B A := inv_match h1\n have h4 : matching (invRel S) C B := inv_match h2\n show fcnl_on (compRel (invRel R) (invRel S)) C from comp_fcnl h4 h3\n done\n done\n done\n\ntheorem Theorem_8_1_3_3 {U V W : Type} {A : Set U} {B : Set V} {C : Set W}\n (h1 : A \u223c B) (h2 : B \u223c C) : A \u223c C := by\n obtain (R : Rel U V) (h3 : matching R A B) from h1\n obtain (S : Rel V W) (h4 : matching S B C) from h2\n apply Exists.intro (compRel S R)\n show matching (compRel S R) A C from comp_match h3 h4\n done\n\nlemma I_def (k n : Nat) : k \u2208 I n \u2194 k < n := by rfl\n\nlemma denum_def {U : Type} (A : Set U) : denum A \u2194 Univ Nat \u223c A := by rfl\n\n/- Section 8.1\u00bd -/\nlemma numElts_def {U : Type} (A : Set U) (n : Nat) :\n numElts A n \u2194 I n \u223c A := by rfl\n\nlemma finite_def {U : Type} (A : Set U) :\n finite A \u2194 \u2203 (n : Nat), numElts A n := by rfl\n\nlemma fcnl_on_empty {U V : Type}\n (R : Rel U V) {A : Set U} (h1 : empty A) : fcnl_on R A := by\n define\n fix a : U\n assume h2 : a \u2208 A --Goal : \u2203! (y : V), R a y\n contradict h1 with h3 --Goal : \u2203 (x : U), x \u2208 A\n show \u2203 (x : U), x \u2208 A from Exists.intro a h2\n done\n\nlemma empty_match {U V : Type} {A : Set U} {B : Set V}\n (h1 : empty A) (h2 : empty B) : matching (emptyRel U V) A B := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix a : U; fix b : V\n assume h3 : emptyRel U V a b --Goal : a \u2208 A \u2227 b \u2208 B\n by_contra h4 --Goal : False\n define at h3\n show False from h3\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on emptyRel\n show fcnl_on (emptyRel U V) A from fcnl_on_empty (emptyRel U V) h1\n done\n \u00b7 -- Proof of fcnl_on (invRel emptyRel)\n show fcnl_on (invRel (emptyRel U V)) B from\n fcnl_on_empty (invRel (emptyRel U V)) h2\n done\n done\n\nlemma I_0_empty : empty (I 0) := by\n define\n by_contra h1 --h1 : \u2203 (x : Nat), x \u2208 I 0\n obtain (x : Nat) (h2 : x \u2208 I 0) from h1\n define at h2 --h2 : x < 0\n linarith\n done\n\ntheorem zero_elts_iff_empty {U : Type} (A : Set U) :\n numElts A 0 \u2194 empty A := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : numElts A 0\n define\n by_contra h2 --h2 : \u2203 (x : U), x \u2208 A\n obtain (x : U) (h3 : x \u2208 A) from h2\n define at h1\n obtain (R : Rel Nat U) (h4 : matching R (I 0) A) from h1\n define at h4\n --h4 : rel_within R (I 0) A \u2227 fcnl_on R (I 0) \u2227 fcnl_on (invRel R) A\n obtain (j : Nat) (h5 : invRel R x j) from\n fcnl_exists h4.right.right h3\n define at h5 --h5 : R j x\n have h6 : j \u2208 I 0 \u2227 x \u2208 A := h4.left h5\n contradict I_0_empty --Goal : \u2203 (x : Nat), x \u2208 I 0\n show \u2203 (x : Nat), x \u2208 I 0 from Exists.intro j h6.left\n done\n \u00b7 -- (\u2190)\n assume h1 : empty A\n show \u2203 (R : Rel Nat U), matching R (I 0) A from\n Exists.intro (emptyRel Nat U) (empty_match I_0_empty h1)\n done\n done\n\ntheorem nonempty_of_pos_numElts {U : Type} {A : Set U} {n : Nat}\n (h1 : numElts A n) (h2 : n > 0) : \u2203 (x : U), x \u2208 A := by\n define at h1\n obtain (R : Rel Nat U) (h3 : matching R (I n) A) from h1\n define at h3\n have h4 : 0 \u2208 I n := h2\n obtain (x : U) (h5 : R 0 x) from fcnl_exists h3.right.left h4\n have h6 : 0 \u2208 I n \u2227 x \u2208 A := h3.left h5\n show \u2203 (x : U), x \u2208 A from Exists.intro x h6.right\n done\n\ntheorem relext {U V : Type} {R S : Rel U V}\n (h : \u2200 (u : U) (v : V), R u v \u2194 S u v) : R = S := by\n have h2 : extension R = extension S := by\n apply Set.ext\n fix (u, v) : U \u00d7 V --Goal : (u, v) \u2208 extension R \u2194 (u, v) \u2208 extension S\n rewrite [ext_def, ext_def] --Goal : R u v \u2194 S u v\n show R u v \u2194 S u v from h u v\n done\n show R = S from\n calc R\n _ = RelFromExt (extension R) := by rfl\n _ = RelFromExt (extension S) := by rw [h2]\n _ = S := by rfl\n done\n\nlemma remove_one_def {U V : Type} (R : Rel U V) (u x : U) (v y : V) :\n remove_one R u v x y \u2194\n x \u2260 u \u2227 y \u2260 v \u2227 (R x y \u2228 (R x v \u2227 R u y)) := by rfl\n\nlemma remove_one_rel_within {U V : Type}\n {R : Rel U V} {A : Set U} {B : Set V} {x u : U} {y v : V}\n (h1 : matching R A B) (h2 : remove_one R u v x y) :\n x \u2208 A \\ {u} \u2227 y \u2208 B \\ {v} := by\n define at h1 --h1 : rel_within R A B \u2227 fcnl_on R A \u2227 fcnl_on (invRel R) B\n define at h2 --h2 : x \u2260 u \u2227 y \u2260 v \u2227 (R x y \u2228 R x v \u2227 R u y)\n have h3 : x \u2208 A \u2227 y \u2208 B := by\n by_cases on h2.right.right with h3\n \u00b7 -- Case 1. h3 : R x y\n show x \u2208 A \u2227 y \u2208 B from h1.left h3\n done\n \u00b7 -- Case 2. h3 : R x v \u2227 R u y\n have h4 : x \u2208 A \u2227 v \u2208 B := h1.left h3.left\n have h5 : u \u2208 A \u2227 y \u2208 B := h1.left h3.right\n show x \u2208 A \u2227 y \u2208 B from And.intro h4.left h5.right\n done\n done\n show x \u2208 A \\ {u} \u2227 y \u2208 B \\ {v} from\n And.intro (And.intro h3.left h2.left) (And.intro h3.right h2.right.left)\n done\n\nlemma remove_one_inv {U V : Type} (R : Rel U V) (u : U) (v : V) :\n invRel (remove_one R u v) = remove_one (invRel R) v u := by\n apply relext\n fix y : V; fix x : U\n --Goal : invRel (remove_one R u v) y x \u2194 remove_one (invRel R) v u y x\n rewrite [invRel_def, remove_one_def, remove_one_def]\n rewrite [invRel_def, invRel_def, invRel_def]\n rewrite [\u2190and_assoc, \u2190and_assoc]\n --Goal : (x \u2260 u \u2227 y \u2260 v) \u2227 (R x y \u2228 R x v \u2227 R u y) \u2194\n -- (y \u2260 v \u2227 x \u2260 u) \u2227 (R x y \u2228 R u y \u2227 R x v)\n have h1 : x \u2260 u \u2227 y \u2260 v \u2194 y \u2260 v \u2227 x \u2260 u := and_comm\n have h2 : R x v \u2227 R u y \u2194 R u y \u2227 R x v := and_comm\n rewrite [h1, h2]\n rfl\n done\n\nlemma remove_one_iff {U V : Type}\n {A : Set U} {B : Set V} {R : Rel U V} (h1 : matching R A B)\n {u : U} (h2 : u \u2208 A) (v : V) {x : U} (h3 : x \u2208 A \\ {u}) :\n \u2203 w \u2208 A, \u2200 (y : V), remove_one R u v x y \u2194 R w y := sorry\n\ntheorem remove_one_fcnl {U V : Type}\n {R : Rel U V} {A : Set U} {B : Set V} {u : U}\n (h1 : matching R A B) (h2 : u \u2208 A) (v : V) :\n fcnl_on (remove_one R u v) (A \\ {u}) := by\n define\n fix x : U\n assume h3 : x \u2208 A \\ {u} --Goal : \u2203! (y : V), remove_one R u v x y\n obtain (w : U) (h4 : w \u2208 A \u2227 \u2200 (y : V),\n remove_one R u v x y \u2194 R w y) from remove_one_iff h1 h2 v h3\n define at h1\n exists_unique\n \u00b7 -- Existence\n obtain (y : V) (h5 : R w y) from fcnl_exists h1.right.left h4.left\n apply Exists.intro y\n rewrite [h4.right]\n show R w y from h5\n done\n \u00b7 -- Uniqueness\n fix y1 : V; fix y2 : V\n rewrite [h4.right, h4.right]\n assume h5 : R w y1\n assume h6 : R w y2\n show y1 = y2 from fcnl_unique h1.right.left h4.left h5 h6\n done\n done\n\ntheorem remove_one_match {U V : Type}\n {R : Rel U V} {A : Set U} {B : Set V} {u : U} {v : V}\n (h1 : matching R A B) (h2 : u \u2208 A) (h3 : v \u2208 B) :\n matching (remove_one R u v) (A \\ {u}) (B \\ {v}) := by\n define\n apply And.intro\n \u00b7 -- Proof of rel_within\n define\n fix x : U; fix y : V\n assume h4 : remove_one R u v x y\n show x \u2208 A \\ {u} \u2227 y \u2208 B \\ {v} from remove_one_rel_within h1 h4\n done\n \u00b7 -- Proof of fcnl_ons\n apply And.intro (remove_one_fcnl h1 h2 v)\n rewrite [remove_one_inv]\n show fcnl_on (remove_one (invRel R) v u) (B \\ {v}) from\n remove_one_fcnl (inv_match h1) h3 u\n done\n\ntheorem remove_one_equinum {U V : Type}\n {A : Set U} {B : Set V} {u : U} {v : V}\n (h1 : A \u223c B) (h2 : u \u2208 A) (h3 : v \u2208 B) : A \\ {u} \u223c B \\ {v} := by\n define\n obtain (R : Rel U V) (h4 : matching R A B) from h1\n apply Exists.intro (remove_one R u v)\n show matching (remove_one R u v) (A \\ {u}) (B \\ {v}) from\n remove_one_match h4 h2 h3\n done\n\n", "theoremStatement": "lemma I_max (n : Nat) : n \u2208 I (n + 1) ", "theoremName": "HTPI.I_max", "fileCreated": {"commit": "16a90bdd1803de01730cb7eb7cfdf6ace669bfbf", "date": "2023-03-26"}, "theoremCreated": {"commit": "01538f73db4cc69cc47bd51e795bf25d251c4b6d", "date": "2023-05-31"}, "file": "htpi/HTPILib/Chap8Part1.lean", "module": "HTPILib.Chap8Part1", "jsonFile": "HTPILib.Chap8Part1.jsonl", "positionMetadata": {"lineInFile": 487, 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"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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n define\n linarith\n done", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 32}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.IntroLean\nnamespace HTPI\n\n/- Definitions -/\ndef even (n : Int) : Prop := \u2203 (k : Int), n = 2 * k\n\ndef odd (n : Int) : Prop := \u2203 (k : Int), n = 2 * k + 1\n\n/- Sections 3.1 and 3.2 -/\ntheorem Example_3_2_4_v2 (P Q R : Prop)\n (h : P \u2192 (Q \u2192 R)) : \u00acR \u2192 (P \u2192 \u00acQ) := by\n assume h2 : \u00acR\n assume h3 : P\n by_contra h4\n have h5 : Q \u2192 R := h h3\n have h6 : R := h5 h4\n show False from h2 h6\n done\n\ntheorem Example_3_2_4_v3 (P Q R : Prop)\n (h : P \u2192 (Q \u2192 R)) : \u00acR \u2192 (P \u2192 \u00acQ) := by\n assume h2 : \u00acR\n assume h3 : P\n by_contra h4\n contradict h2\n show R from h h3 h4\n done\n\ntheorem Like_Example_3_2_5\n (U : Type) (A B C : Set U) (a : U)\n (h1 : a \u2208 A) (h2 : a \u2209 A \\ B)\n (h3 : a \u2208 B \u2192 a \u2208 C) : a \u2208 C := by\n apply h3 _\n define at h2\n demorgan at h2; conditional at h2\n show a \u2208 B from h2 h1\n done\n\n/- Section 3.3 -/\nexample (U : Type) (P Q : Pred U)\n (h1 : \u2200 (x : U), P x \u2192 \u00acQ x)\n (h2 : \u2200 (x : U), Q x) :\n \u00ac\u2203 (x : U), P x := by\n quant_neg --Goal is now \u2200 (x : U), \u00acP x\n fix y : U\n have h3 : P y \u2192 \u00acQ y := h1 y\n have h4 : Q y := h2 y\n contrapos at h3 --Now h3 : Q y \u2192 \u00acP y\n show \u00acP y from h3 h4\n done\n\nexample (U : Type) (A B C : Set U) (h1 : A \u2286 B \u222a C)\n (h2 : \u2200 (x : U), x \u2208 A \u2192 x \u2209 B) : A \u2286 C := by\n define --Goal : \u2200 \u2983a : U\u2984, a \u2208 A \u2192 a \u2208 C\n fix y : U\n assume h3 : y \u2208 A\n have h4 : y \u2209 B := h2 y h3\n define at h1 --h1 : \u2200 \u2983a : U\u2984, a \u2208 A \u2192 a \u2208 B \u222a C\n have h5 : y \u2208 B \u222a C := h1 h3\n define at h5 --h5 : y \u2208 B \u2228 y \u2208 C\n conditional at h5 --h5 : \u00acy \u2208 B \u2192 y \u2208 C\n show y \u2208 C from h5 h4\n done\n\nexample (U : Type) (P Q : Pred U)\n (h1 : \u2200 (x : U), \u2203 (y : U), P x \u2192 \u00ac Q y)\n (h2 : \u2203 (x : U), \u2200 (y : U), P x \u2192 Q y) :\n \u2203 (x : U), \u00acP x := by\n obtain (a : U)\n (h3 : \u2200 (y : U), P a \u2192 Q y) from h2\n have h4 : \u2203 (y : U), P a \u2192 \u00ac Q y := h1 a\n obtain (b : U) (h5 : P a \u2192 \u00ac Q b) from h4\n have h6 : P a \u2192 Q b := h3 b\n apply Exists.intro a _\n by_contra h7\n show False from h5 h7 (h6 h7)\n done\n\ntheorem Example_3_3_5 (U : Type) (B : Set U)\n (F : Set (Set U)) : \u22c3\u2080 F \u2286 B \u2192 F \u2286 \ud835\udcab B := by\n assume h1 : \u22c3\u2080 F \u2286 B\n define\n fix x : Set U\n assume h2 : x \u2208 F\n define\n fix y : U\n assume h3 : y \u2208 x\n define at h1\n apply h1 _\n define\n apply Exists.intro x _\n show x \u2208 F \u2227 y \u2208 x from And.intro h2 h3\n done\n\n/- Section 3.4 -/\ntheorem Like_Example_3_4_1 (U : Type)\n (A B C D : Set U) (h1 : A \u2286 B)\n (h2 : \u00ac\u2203 (c : U), c \u2208 C \u2229 D) :\n A \u2229 C \u2286 B \\ D := by\n define\n fix x : U\n assume h3 : x \u2208 A \u2229 C\n define at h3; define\n apply And.intro\n \u00b7 -- Proof that x \u2208 B.\n show x \u2208 B from h1 h3.left\n done\n \u00b7 -- Proof that x \u2209 D.\n contradict h2 with h4\n apply Exists.intro x\n show x \u2208 C \u2229 D from And.intro h3.right h4\n done\n done\n\nexample (U : Type) (P Q : Pred U)\n (h1 : \u2200 (x : U), P x \u2194 Q x) :\n (\u2203 (x : U), P x) \u2194 \u2203 (x : U), Q x := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : \u2203 (x : U), P x\n obtain (u : U) (h3 : P u) from h2\n have h4 : P u \u2194 Q u := h1 u\n apply Exists.intro u\n show Q u from h4.ltr h3\n done\n \u00b7 -- (\u2190)\n assume h2 : \u2203 (x : U), Q x\n obtain (u : U) (h3 : Q u) from h2\n show \u2203 (x : U), P x from Exists.intro u ((h1 u).rtl h3)\n done\n done\n\ntheorem Example_3_4_5 (U : Type)\n (A B C : Set U) : A \u2229 (B \\ C) = (A \u2229 B) \\ C := by\n apply Set.ext\n fix x : U\n show x \u2208 A \u2229 (B \\ C) \u2194 x \u2208 (A \u2229 B) \\ C from\n calc x \u2208 A \u2229 (B \\ C)\n _ \u2194 x \u2208 A \u2227 (x \u2208 B \u2227 x \u2209 C) := Iff.refl _\n _ \u2194 (x \u2208 A \u2227 x \u2208 B) \u2227 x \u2209 C := and_assoc.symm\n _ \u2194 x \u2208 (A \u2229 B) \\ C := Iff.refl _\n done\n\n/- Section 3.5 -/\ntheorem Example_3_5_2\n (U : Type) (A B C : Set U) :\n A \\ (B \\ C) \u2286 (A \\ B) \u222a C := by\n fix x : U\n assume h1 : x \u2208 A \\ (B \\ C)\n define; define at h1\n have h2 : x \u2209 B \\ C := h1.right\n define at h2; demorgan at h2\n --h2 : x \u2209 B \u2228 x \u2208 C\n by_cases on h2\n \u00b7 -- Case 1. h2 : x \u2209 B\n apply Or.inl\n show x \u2208 A \\ B from And.intro h1.left h2\n done\n \u00b7 -- Case 2. h2 : x \u2208 C\n apply Or.inr\n show x \u2208 C from h2\n done\n done\n\nexample (U : Type) (A B C : Set U)\n (h1 : A \\ B \u2286 C) : A \u2286 B \u222a C := by\n fix x : U\n assume h2 : x \u2208 A\n define\n or_right with h3\n show x \u2208 C from h1 (And.intro h2 h3)\n done\n\nexample\n (U : Type) (A B C : Set U) (h1 : A \u2286 B \u222a C)\n (h2 : \u00ac\u2203 (x : U), x \u2208 A \u2229 B) : A \u2286 C := by\n fix a : U\n assume h3 : a \u2208 A\n quant_neg at h2\n have h4 : a \u2208 B \u222a C := h1 h3\n have h5 : a \u2209 A \u2229 B := h2 a\n define at h4\n define at h5; demorgan at h5\n disj_syll h5 h3 --h5 : \u00aca \u2208 B\n disj_syll h4 h5 --h4 : a \u2208 C\n show a \u2208 C from h4\n done\n\nexample\n (U : Type) (A B C : Set U) (h1 : A \u2286 B \u222a C)\n (h2 : \u00ac\u2203 (x : U), x \u2208 A \u2229 B) : A \u2286 C := by\n fix a : U\n assume h3 : a \u2208 A\n have h4 : a \u2208 B \u222a C := h1 h3\n define at h4\n have h5 : a \u2209 B := by\n contradict h2 with h6\n show \u2203 (x : U), x \u2208 A \u2229 B from\n Exists.intro a (And.intro h3 h6)\n done\n disj_syll h4 h5 --h4 : a \u2208 C\n show a \u2208 C from h4\n done\n\n/- Section 3.6 -/\n", "theoremStatement": "theorem empty_union {U : Type} (B : Set U) :\n \u2205 \u222a B = B ", "theoremName": "HTPI.empty_union", "fileCreated": {"commit": 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"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.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.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "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.Meta.Tactic.Delta", "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.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "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.Meta.Tactic.Symm", "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.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "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", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply Set.ext\n fix x : U\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : x \u2208 \u2205 \u222a B\n define at h1\n have h2 : x \u2209 \u2205 := by\n by_contra h3\n define at h3 --h3 : False\n show False from h3\n done\n disj_syll h1 h2 --h1 : x \u2208 B\n show x \u2208 B from h1\n done\n \u00b7 -- (\u2190)\n assume h1 : x \u2208 B\n show x \u2208 \u2205 \u222a B from Or.inr h1\n done\n done", "proofType": "tactic", "proofLengthLines": 19, "proofLengthTokens": 368}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap5\nnamespace HTPI\n\n/- Definitions -/\ndef invRel {U V : Type} (R : Rel U V) : Rel V U :=\n RelFromExt (inv (extension R))\n\ndef rel_within {U V : Type} (R : Rel U V) (A : Set U) (B : Set V) : Prop :=\n \u2200 \u2983x : U\u2984 \u2983y : V\u2984, R x y \u2192 x \u2208 A \u2227 y \u2208 B\n\ndef fcnl_on {U V : Type} (R : Rel U V) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203! (y : V), R x y\n\ndef matching {U V : Type} (R : Rel U V) (A : Set U) (B : Set V) : Prop :=\n rel_within R A B \u2227 fcnl_on R A \u2227 fcnl_on (invRel R) B\n\ndef equinum {U V : Type} (A : Set U) (B : Set V) : Prop :=\n \u2203 (R : Rel U V), matching R A B\n\nnotation:50 A:50 \" \u223c \" B:50 => equinum A B\n\ndef RelWithinFromFunc {U V : Type} (f : U \u2192 V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 f x = y\n\ndef one_one_on {U V : Type} (f : U \u2192 V) (A : Set U) : Prop :=\n \u2200 \u2983x1 x2 : U\u2984, x1 \u2208 A \u2192 x2 \u2208 A \u2192 f x1 = f x2 \u2192 x1 = x2\n\ndef Univ (U : Type) : Set U := {x : U | True}\n\ndef compRel {U V W : Type} (S : Rel V W) (R : Rel U V) : Rel U W :=\n RelFromExt (comp (extension S) (extension R))\n\ndef I (n : Nat) : Set Nat := {k : Nat | k < n}\n\ndef finite {U : Type} (A : Set U) : Prop :=\n \u2203 (n : Nat), I n \u223c A\n\ndef denum {U : Type} (A : Set U) : Prop :=\n Univ Nat \u223c A\n\ndef ctble {U : Type} (A : Set U) : Prop :=\n finite A \u2228 denum A\n\ndef numElts {U : Type} (A : Set U) (n : Nat) : Prop := I n \u223c A\n\ndef emptyRel (U V : Type) (x : U) (y : V) : Prop := False\n\ndef remove_one {U V : Type} (R : Rel U V) (u : U) (v : V)\n (x : U) (y : V) : Prop := x \u2260 u \u2227 y \u2260 v \u2227 (R x y \u2228 (R x v \u2227 R u y))\n\ndef one_match {U V : Type} (a : U) (b : V)\n (x : U) (y : V) : Prop := x = a \u2227 y = b\n\n/- Section 8.1 -/\nlemma invRel_def {U V : Type} (R : Rel U V) (u : U) (v : V) :\n invRel R v u \u2194 R u v := by rfl\n\ntheorem equinum_image {U V : Type} {A : Set U} {B : Set V} {f : U \u2192 V}\n (h1 : one_one_on f A) (h2 : image f A = B) : A \u223c B := by\n rewrite [\u2190h2]\n define --Goal : \u2203 (R : Rel U V), matching R A (image f A)\n set R : Rel U V := RelWithinFromFunc f A\n apply Exists.intro R\n define --Goal : rel_within R A (image f A) \u2227\n --fcnl_on R A \u2227 fcnl_on (invRel R) (image f A)\n apply And.intro\n \u00b7 -- Proof of rel_within\n define --Goal : \u2200 \u2983x : U\u2984 \u2983y : V\u2984, R x y \u2192 x \u2208 A \u2227 y \u2208 image f A\n fix x : U; fix y : V\n assume h3 : R x y --Goal : x \u2208 A \u2227 y \u2208 image f A\n define at h3 --h3 : x \u2208 A \u2227 f x = y\n apply And.intro h3.left\n define\n show \u2203 x \u2208 A, f x = y from Exists.intro x h3\n done\n \u00b7 -- Proofs of fcnl_ons\n apply And.intro\n \u00b7 -- Proof of fcnl_on R A\n define --Goal : \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203! (y : V), R x y\n fix x : U\n assume h3 : x \u2208 A\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (f x)\n define --Goal : x \u2208 A \u2227 f x = f x\n apply And.intro h3\n rfl\n done\n \u00b7 -- Uniqueness\n fix y1 : V; fix y2 : V\n assume h4 : R x y1\n assume h5 : R x y2 --Goal : y1 = y2\n define at h4; define at h5\n --h4 : x \u2208 A \u2227 f x = y1; h5 : x \u2208 A \u2227 f x = y2\n rewrite [h4.right] at h5\n show y1 = y2 from h5.right\n done\n done\n \u00b7 -- Proof of fcnl_on (invRel R) (image f A)\n define --Goal : \u2200 \u2983x : V\u2984, x \u2208 image f A \u2192 \u2203! (y : U), invRel R x y\n fix y : V\n assume h3 : y \u2208 image f A\n obtain (x : U) (h4 : x \u2208 A \u2227 f x = y) from h3\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro x\n define\n show x \u2208 A \u2227 f x = y from h4\n done\n \u00b7 -- Uniqueness\n fix x1 : U; fix x2 : U\n assume h5 : invRel R y x1\n assume h6 : invRel R y x2\n define at h5; define at h6\n --h5 : x1 \u2208 A \u2227 f x1 = y; h6 : x2 \u2208 A \u2227 f x2 = y\n rewrite [\u2190h6.right] at h5\n show x1 = x2 from h1 h5.left h6.left h5.right\n done\n done\n done\n done\n\nlemma id_one_one_on {U : Type} (A : Set U) : one_one_on id A := by\n define\n fix x1 : U; fix x2 : U\n assume h1 : x1 \u2208 A\n assume h2 : x2 \u2208 A\n assume h3 : id x1 = id x2\n show x1 = x2 from h3\n done\n\nlemma image_id {U : Type} (A : Set U) : image id A = A := by\n apply Set.ext\n fix x : U\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : x \u2208 image id A\n obtain (y : U) (h2 : y \u2208 A \u2227 id y = x) from h1\n rewrite [\u2190h2.right]\n show id y \u2208 A from h2.left\n done\n \u00b7 -- (\u2190)\n assume h1 : x \u2208 A\n apply Exists.intro x --Goal : x \u2208 A \u2227 id x = x\n apply And.intro h1\n rfl\n done\n done\n\ntheorem Theorem_8_1_3_1 {U : Type} (A : Set U) : A \u223c A :=\n equinum_image (id_one_one_on A) (image_id A)\n\nlemma inv_inv {U V : Type} (R : Rel U V) : invRel (invRel R) = R := by rfl\n\nlemma inv_match {U V : Type} {R : Rel U V} {A : Set U} {B : Set V}\n (h : matching R A B) : matching (invRel R) B A := by\n define --Goal : rel_within (invRel R) B A \u2227\n --fcnl_on (invRel R) B \u2227 fcnl_on (invRel (invRel R)) A\n define at h --h : rel_within R A B \u2227 fcnl_on R A \u2227 fcnl_on (invRel R) B\n apply And.intro\n \u00b7 -- Proof that rel_within (invRel R) B A\n define --Goal : \u2200 \u2983x : V\u2984 \u2983y : U\u2984, invRel R x y \u2192 x \u2208 B \u2227 y \u2208 A\n fix y : V; fix x : U\n assume h1 : invRel R y x\n define at h1 --h1 : R x y\n have h2 : x \u2208 A \u2227 y \u2208 B := h.left h1\n show y \u2208 B \u2227 x \u2208 A from And.intro h2.right h2.left\n done\n \u00b7 -- proof that fcnl_on (inv R) B \u2227 fcnl_on (inv (inv R)) A\n rewrite [inv_inv]\n show fcnl_on (invRel R) B \u2227 fcnl_on R A from\n And.intro h.right.right h.right.left\n done\n done\n\n", "theoremStatement": "theorem Theorem_8_1_3_2 {U V : Type} {A : Set U} {B : Set V}\n (h : A \u223c B) : B \u223c A ", "theoremName": "HTPI.Theorem_8_1_3_2", "fileCreated": {"commit": "16a90bdd1803de01730cb7eb7cfdf6ace669bfbf", "date": "2023-03-26"}, "theoremCreated": {"commit": "fe2721d65e0771872fe4cb8117517e643ac1dc1f", "date": "2023-05-10"}, "file": "htpi/HTPILib/Chap8Part1.lean", "module": "HTPILib.Chap8Part1", "jsonFile": "HTPILib.Chap8Part1.jsonl", "positionMetadata": {"lineInFile": 178, "tokenPositionInFile": 5452, "theoremPositionInFile": 25}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, 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"Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "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", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "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", "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", "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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain (R : Rel U V) (h1 : matching R A B) from h\n apply Exists.intro (invRel R)\n show matching (invRel R) B A from inv_match h1\n done", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 145}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap7\nnamespace HTPI\n\n/- Definitions -/\ndef fnz (n : Nat) : Int := if 2 \u2223 n then \u2191(n / 2) else -\u2191((n + 1) / 2)\n\ndef fzn (a : Int) : Nat := if a \u2265 0 then 2 * Int.toNat a else 2 * Int.toNat (-a) - 1\n\ndef tri (k : Nat) : Nat := k * (k + 1) / 2\n\ndef fnnn (p : Nat \u00d7 Nat) : Nat := tri (p.1 + p.2) + p.1\n\ndef num_elts_below (A : Set Nat) (m s : Nat) : Prop :=\n match m with\n | 0 => s = 0\n | n + 1 => (n \u2208 A \u2227 1 \u2264 s \u2227 num_elts_below A n (s - 1)) \u2228\n (n \u2209 A \u2227 num_elts_below A n s)\n\ndef enum (A : Set Nat) (s n : Nat) : Prop := n \u2208 A \u2227 num_elts_below A n s\n\ndef unique_val_on_N {U : Type} (R : Rel Nat U) : Prop :=\n \u2200 \u2983n : Nat\u2984 \u2983x1 x2 : U\u2984, R n x1 \u2192 R n x2 \u2192 x1 = x2\n\ndef nat_rel_onto {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n \u2200 \u2983x : U\u2984, x \u2208 A \u2192 \u2203 (n : Nat), R n x\n\ndef fcnl_onto_from_nat {U : Type} (R : Rel Nat U) (A : Set U) : Prop :=\n unique_val_on_N R \u2227 nat_rel_onto R A\n\ndef fcnl_one_one_to_nat {U : Type} (R : Rel U Nat) (A : Set U) : Prop :=\n fcnl_on R A \u2227 \u2200 \u2983x1 x2 : U\u2984 \u2983n : Nat\u2984,\n (x1 \u2208 A \u2227 R x1 n) \u2192 (x2 \u2208 A \u2227 R x2 n) \u2192 x1 = x2\n\ndef least_rel_to {U : Type} (S : Rel Nat U) (x : U) (n : Nat) : Prop :=\n S n x \u2227 \u2200 (m : Nat), S m x \u2192 n \u2264 m\n\ndef restrict_to {U V : Type} (S : Rel U V) (A : Set U)\n (x : U) (y : V) : Prop := x \u2208 A \u2227 S x y\n\ndef fqn (q : Rat) : Nat := fnnn (fzn q.num, q.den)\n\ndef Set_rp_below (m : Nat) : Set Nat := {n : Nat | rel_prime m n \u2227 n < m}\n\ndef Set_prod {U V : Type} (A : Set U) (B : Set V) : Set (U \u00d7 V) :=\n {(a, b) : U \u00d7 V | a \u2208 A \u2227 b \u2208 B}\n\nnotation:75 A:75 \" \u00d7\u209b \" B:75 => Set_prod A B\n\ndef Rel_prod {U V W X : Type} (R : Rel U V) (S : Rel W X)\n (p : U \u00d7 W) (q : V \u00d7 X) : Prop := R p.1 q.1 \u2227 S p.2 q.2\n\nnotation:75 R:75 \" \u00d7\u1d63 \" S:75 => Rel_prod R S\n\ndef qr (n a : Nat) : Nat \u00d7 Nat := (a / n, a % n)\n\ndef mod_mod (m n a : Nat) : Nat \u00d7 Nat := (a % m, a % n)\n\ndef enum_union_fam {U : Type}\n (F : Set (Set U)) (f : Set U \u2192 Rel Nat U) (R : Rel Nat (Set U))\n (n : Nat) (a : U) : Prop := \u2203 (p : Nat \u00d7 Nat), fnnn p = n \u2227\n \u2203 A \u2208 F, R p.1 A \u2227 (f A) p.2 a\n\ndef seq {U : Type} (A : Set U) : Set (List U) :=\n {l : List U | \u2200 x \u2208 l, x \u2208 A}\n\ndef seq_by_length {U : Type} (A : Set U) (n : Nat) : Set (List U) :=\n {l : List U | l \u2208 seq A \u2227 l.length = n}\n\ndef seq_cons (U : Type) (p : U \u00d7 (List U)) : List U := p.1 :: p.2\n\ndef sbl_set {U : Type} (A : Set U) : Set (Set (List U)) :=\n {S : Set (List U) | \u2203 (n : Nat), seq_by_length A n = S}\n\ndef rep_common_image\n {U V : Type} (R S : Rel U V) (X0 : Set U) (n : Nat) : Set U :=\n match n with\n | 0 => X0\n | m + 1 => {a : U | \u2203 x \u2208 rep_common_image R S X0 m,\n \u2203 (y : V), R x y \u2227 S a y}\n\ndef cum_rep_image {U V : Type} (R S : Rel U V) (X0 : Set U) : Set U :=\n {a : U | \u2203 (n : Nat), a \u2208 rep_common_image R S X0 n}\n\ndef csb_match {U V : Type} (R S : Rel U V) (X0 : Set U)\n (x : U) (y : V) : Prop := x \u2208 cum_rep_image R S X0 \u2227 R x y \u2228\n x \u2209 cum_rep_image R S X0 \u2227 S x y\n\n/- Section 8.1 -/\n#eval [fnz 0, fnz 1, fnz 2, fnz 3, fnz 4, fnz 5, fnz 6]\n --Answer: [0, -1, 1, -2, 2, -3, 3]\n\n#eval [fzn 0, fzn (-1), fzn 1, fzn (-2), fzn 2, fzn (-3), fzn 3]\n --Answer: [0, 1, 2, 3, 4, 5, 6]\n\nlemma fnz_even (k : Nat) : fnz (2 * k) = \u2191k := by\n have h1 : 2 \u2223 2 * k := by\n apply Exists.intro k\n rfl\n done\n have h2 : fnz (2 * k) = if 2 \u2223 2 * k then \u2191(2 * k / 2)\n else -\u2191((2 * k + 1) / 2) := by rfl\n rewrite [if_pos h1] at h2 --h2 : fnz (2 * k) = \u2191(2 * k / 2)\n have h3 : 0 < 2 := by linarith\n rewrite [Nat.mul_div_cancel_left k h3] at h2\n show fnz (2 * k) = \u2191k from h2\n done\n\nlemma fnz_odd (k : Nat) : fnz (2 * k + 1) = -\u2191(k + 1) := sorry\n\nlemma fzn_nat (k : Nat) : fzn \u2191k = 2 * k := by rfl\n\nlemma fzn_neg_succ_nat (k : Nat) : fzn (-\u2191(k + 1)) = 2 * k + 1 := by rfl\n\n--From exercises of Section 6.1\ntheorem Exercise_6_1_16a1 : \u2200 (n : Nat), nat_even n \u2228 nat_odd n := sorry\n\nlemma fzn_fnz : fzn \u2218 fnz = id := by\n apply funext --Goal : \u2200 (x : Nat), (fzn \u2218 fnz) x = id x\n fix n : Nat\n rewrite [comp_def] --Goal : fzn (fnz n) = id n\n have h1 : nat_even n \u2228 nat_odd n := Exercise_6_1_16a1 n\n by_cases on h1\n \u00b7 -- Case 1. h1 : nat_even n\n obtain (k : Nat) (h2 : n = 2 * k) from h1\n rewrite [h2, fnz_even, fzn_nat]\n rfl\n done\n \u00b7 -- Case 2. h1 : nat_odd n\n obtain (k : Nat) (h2 : n = 2 * k + 1) from h1\n rewrite [h2, fnz_odd, fzn_neg_succ_nat]\n rfl\n done\n done\n\nlemma fnz_fzn : fnz \u2218 fzn = id := sorry\n\nlemma fzn_one_one : one_to_one fzn := Theorem_5_3_3_1 fzn fnz fnz_fzn\n\nlemma fzn_onto : onto fzn := Theorem_5_3_3_2 fzn fnz fzn_fnz\n\nlemma fnz_one_one : one_to_one fnz := Theorem_5_3_3_1 fnz fzn fzn_fnz\n\nlemma fnz_onto : onto fnz := Theorem_5_3_3_2 fnz fzn fnz_fzn\n\n", "theoremStatement": "lemma fnnn_def (a b : Nat) : fnnn (a, b) = tri (a + b) + a ", "theoremName": "HTPI.fnnn_def", "fileCreated": {"commit": "fe2721d65e0771872fe4cb8117517e643ac1dc1f", "date": "2023-05-10"}, "theoremCreated": {"commit": "01538f73db4cc69cc47bd51e795bf25d251c4b6d", "date": "2023-05-31"}, "file": "htpi/HTPILib/Chap8Part2.lean", "module": "HTPILib.Chap8Part2", "jsonFile": "HTPILib.Chap8Part2.jsonl", "positionMetadata": {"lineInFile": 147, "tokenPositionInFile": 4695, "theoremPositionInFile": 39}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 9, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", 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"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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6", "HTPILib.Chap7"]}, "proofMetadata": {"hasProof": true, "proof": ":= by rfl", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 9}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport HTPILib.Chap6\nnamespace HTPI\n\n/- Definitions -/\nlemma mod_succ_lt (a n : Nat) : a % (n + 1) < n + 1 := by\n have h : n + 1 > 0 := Nat.succ_pos n\n show a % (n + 1) < n + 1 from Nat.mod_lt a h\n done\n\ndef gcd (a b : Nat) : Nat :=\n match b with\n | 0 => a\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd (n + 1) (a % (n + 1))\n termination_by b\n\nmutual\n def gcd_c1 (a b : Nat) : Int :=\n match b with\n | 0 => 1\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c2 (n + 1) (a % (n + 1))\n --Corresponds to s = t'\n termination_by b\n\n def gcd_c2 (a b : Nat) : Int :=\n match b with\n | 0 => 0\n | n + 1 =>\n have : a % (n + 1) < n + 1 := mod_succ_lt a n\n gcd_c1 (n + 1) (a % (n + 1)) -\n (gcd_c2 (n + 1) (a % (n + 1))) * \u2191(a / (n + 1))\n --Corresponds to t = s' - t'q\n termination_by b\nend\n\ndef prime (n : Nat) : Prop :=\n 2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n\n\ndef prime_factor (p n : Nat) : Prop := prime p \u2227 p \u2223 n\n\ndef all_prime (l : List Nat) : Prop := \u2200 p \u2208 l, prime p\n\ndef nondec (l : List Nat) : Prop :=\n match l with\n | [] => True --Of course, True is a proposition that is always true\n | n :: L => (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L\n\ndef nondec_prime_list (l : List Nat) : Prop := all_prime l \u2227 nondec l\n\ndef prod (l : List Nat) : Nat :=\n match l with\n | [] => 1\n | n :: L => n * (prod L)\n\ndef prime_factorization (n : Nat) (l : List Nat) : Prop :=\n nondec_prime_list l \u2227 prod l = n\n\ndef rel_prime (a b : Nat) : Prop := gcd a b = 1\n\ndef congr_mod (m : Nat) (a b : Int) : Prop := (\u2191m : Int) \u2223 (a - b)\n\ndef cc (m : Nat) (a : Int) : ZMod m := (\u2191a : ZMod m)\n\nnotation:50 a \" \u2261 \" b \" (MOD \" m \")\" => congr_mod m a b\n\nnotation:max \"[\"a\"]_\"m:max => cc m a\n\ndef invertible {m : Nat} (X : ZMod m) : Prop :=\n \u2203 (Y : ZMod m), X * Y = [1]_m\n\ndef num_rp_below (m k : Nat) : Nat :=\n match k with\n | 0 => 0\n | j + 1 => if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j\n\ndef phi (m : Nat) : Nat := num_rp_below m m\n\ndef prod_seq {m : Nat}\n (j k : Nat) (f : Nat \u2192 ZMod m) : ZMod m :=\n match j with\n | 0 => [1]_m\n | n + 1 => prod_seq n k f * f (k + n)\n\ndef maps_below (n : Nat) (g : Nat \u2192 Nat) : Prop := \u2200 i < n, g i < n\n\ndef one_one_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 i1 < n, \u2200 i2 < n, g i1 = g i2 \u2192 i1 = i2\n\ndef onto_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n \u2200 k < n, \u2203 i < n, g i = k\n\ndef perm_below (n : Nat) (g : Nat \u2192 Nat) : Prop :=\n maps_below n g \u2227 one_one_below n g \u2227 onto_below n g\n\ndef inv_mod (m a : Nat) : Nat := Int.toNat ((gcd_c2 m a) % m)\n\ndef swap (u v i : Nat) : Nat :=\n if i = u then v else if i = v then u else i\n\nnamespace Euler --For definitions specific to Euler's theorem\n\ndef F (m i : Nat) : ZMod m := if gcd m i = 1 then [i]_m else [1]_m\n\ndef G (m a i : Nat) : Nat := (a * i) % m\n\ndef Ginv (m a i : Nat) : Nat := G m (inv_mod m a) i\n\nend Euler\n\n/- Section 7.1 -/\ntheorem dvd_mod_of_dvd_a_b {a b d : Nat}\n (h1 : d \u2223 a) (h2 : d \u2223 b) : d \u2223 (a % b) := by\n set q : Nat := a / b\n have h3 : b * q + a % b = a := Nat.div_add_mod a b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n define --Goal : \u2203 (c : Nat), a % b = d * c\n apply Exists.intro (j - k * q)\n show a % b = d * (j - k * q) from\n calc a % b\n _ = b * q + a % b - b * q := (Nat.add_sub_cancel_left _ _).symm\n _ = a - b * q := by rw [h3]\n _ = d * j - d * (k * q) := by rw [h4, h5, mul_assoc]\n _ = d * (j - k * q) := (Nat.mul_sub_left_distrib _ _ _).symm\n done\n\ntheorem dvd_a_of_dvd_b_mod {a b d : Nat}\n (h1 : d \u2223 b) (h2 : d \u2223 (a % b)) : d \u2223 a := sorry\n\n#eval gcd 672 161 --Answer: 7\n\nlemma gcd_base (a : Nat) : gcd a 0 = a := by rfl\n\nlemma gcd_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd a b = gcd b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2] --Goal : gcd a (n + 1) = gcd (n + 1) (a % (n + 1))\n rfl\n done\n\nlemma mod_nonzero_lt (a : Nat) {b : Nat} (h : b \u2260 0) : a % b < b := by\n have h1 : b > 0 := Nat.pos_of_ne_zero h\n show a % b < b from Nat.mod_lt a h1\n done\n\nlemma dvd_self (n : Nat) : n \u2223 n := by\n apply Exists.intro 1\n ring\n done\n\ntheorem gcd_dvd : \u2200 (b a : Nat), (gcd a b) \u2223 a \u2227 (gcd a b) \u2223 b := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat), (gcd a b_1) \u2223 a \u2227 (gcd a b_1) \u2223 b_1\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_base] --Goal: a \u2223 a \u2227 a \u2223 0\n apply And.intro (dvd_self a)\n define\n apply Exists.intro 0\n rfl\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_nonzero a h1]\n --Goal : gcd b (a % b) \u2223 a \u2227 gcd b (a % b) \u2223 b\n have h2 : a % b < b := mod_nonzero_lt a h1\n have h3 : (gcd b (a % b)) \u2223 b \u2227 (gcd b (a % b)) \u2223 (a % b) :=\n ih (a % b) h2 b\n apply And.intro _ h3.left\n show (gcd b (a % b)) \u2223 a from dvd_a_of_dvd_b_mod h3.left h3.right\n done\n done\n\ntheorem gcd_dvd_left (a b : Nat) : (gcd a b) \u2223 a := (gcd_dvd b a).left\n\ntheorem gcd_dvd_right (a b : Nat) : (gcd a b) \u2223 b := (gcd_dvd b a).right\n\nlemma gcd_c1_base (a : Nat) : gcd_c1 a 0 = 1 := by rfl\n\nlemma gcd_c1_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c1 a b = gcd_c2 b (a % b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\nlemma gcd_c2_base (a : Nat) : gcd_c2 a 0 = 0 := by rfl\n\nlemma gcd_c2_nonzero (a : Nat) {b : Nat} (h : b \u2260 0) :\n gcd_c2 a b = gcd_c1 b (a % b) - (gcd_c2 b (a % b)) * \u2191(a / b) := by\n obtain (n : Nat) (h2 : b = n + 1) from exists_eq_add_one_of_ne_zero h\n rewrite [h2]\n rfl\n done\n\ntheorem gcd_lin_comb : \u2200 (b a : Nat),\n (gcd_c1 a b) * \u2191a + (gcd_c2 a b) * \u2191b = \u2191(gcd a b) := by\n by_strong_induc\n fix b : Nat\n assume ih : \u2200 b_1 < b, \u2200 (a : Nat),\n (gcd_c1 a b_1) * \u2191a + (gcd_c2 a b_1) * \u2191b_1 = \u2191(gcd a b_1)\n fix a : Nat\n by_cases h1 : b = 0\n \u00b7 -- Case 1. h1 : b = 0\n rewrite [h1, gcd_c1_base, gcd_c2_base, gcd_base]\n --Goal : 1 * \u2191a + 0 * \u21910 = \u2191a\n ring\n done\n \u00b7 -- Case 2. h1 : b \u2260 0\n rewrite [gcd_c1_nonzero a h1, gcd_c2_nonzero a h1, gcd_nonzero a h1]\n --Goal : gcd_c2 b (a % b) * \u2191a +\n -- (gcd_c1 b (a % b) - gcd_c2 b (a % b) * \u2191(a / b)) * \u2191b =\n -- \u2191(gcd b (a % b))\n set r : Nat := a % b\n set q : Nat := a / b\n set s : Int := gcd_c1 b r\n set t : Int := gcd_c2 b r\n --Goal : t * \u2191a + (s - t * \u2191q) * \u2191b = \u2191(gcd b r)\n have h2 : r < b := mod_nonzero_lt a h1\n have h3 : s * \u2191b + t * \u2191r = \u2191(gcd b r) := ih r h2 b\n have h4 : b * q + r = a := Nat.div_add_mod a b\n rewrite [\u2190h3, \u2190h4]\n rewrite [Nat.cast_add, Nat.cast_mul]\n --Goal : t * (\u2191b * \u2191q + \u2191r) + (s - t * \u2191q) * \u2191b = s * \u2191b + t * \u2191r\n ring\n done\n done\n\n#eval gcd_c1 672 161 --Answer: 6\n#eval gcd_c2 672 161 --Answer: -25\n --Note 6 * 672 - 25 * 161 = 4032 - 4025 = 7 = gcd 672 161\n\ntheorem Theorem_7_1_6 {d a b : Nat} (h1 : d \u2223 a) (h2 : d \u2223 b) :\n d \u2223 gcd a b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191d \u2223 \u2191(gcd a b)\n set s : Int := gcd_c1 a b\n set t : Int := gcd_c2 a b\n have h3 : s * \u2191a + t * \u2191b = \u2191(gcd a b) := gcd_lin_comb b a\n rewrite [\u2190h3] --Goal : \u2191d \u2223 s * \u2191a + t * \u2191b\n obtain (j : Nat) (h4 : a = d * j) from h1\n obtain (k : Nat) (h5 : b = d * k) from h2\n rewrite [h4, h5, Nat.cast_mul, Nat.cast_mul]\n --Goal : \u2191d \u2223 s * (\u2191d * \u2191j) + t * (\u2191d * \u2191k)\n define\n apply Exists.intro (s * \u2191j + t * \u2191k)\n ring\n done\n\n/- Section 7.2 -/\ntheorem dvd_trans {a b c : Nat} (h1 : a \u2223 b) (h2 : b \u2223 c) : a \u2223 c := by\n define at h1; define at h2; define\n obtain (m : Nat) (h3 : b = a * m) from h1\n obtain (n : Nat) (h4 : c = b * n) from h2\n rewrite [h3, mul_assoc] at h4\n apply Exists.intro (m * n)\n show c = a * (m * n) from h4\n done\n\nlemma exists_prime_factor : \u2200 (n : Nat), 2 \u2264 n \u2192\n \u2203 (p : Nat), prime_factor p n := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, 2 \u2264 n_1 \u2192 \u2203 (p : Nat), prime_factor p n_1\n assume h1 : 2 \u2264 n\n by_cases h2 : prime n\n \u00b7 -- Case 1. h2 : prime n\n apply Exists.intro n\n define --Goal : prime n \u2227 n \u2223 n\n show prime n \u2227 n \u2223 n from And.intro h2 (dvd_self n)\n done\n \u00b7 -- Case 2. h2 : \u00acprime n\n define at h2\n --h2 : \u00ac(2 \u2264 n \u2227 \u00ac\u2203 (a b : Nat), a * b = n \u2227 a < n \u2227 b < n)\n demorgan at h2\n disj_syll h2 h1\n obtain (a : Nat) (h3 : \u2203 (b : Nat), a * b = n \u2227 a < n \u2227 b < n) from h2\n obtain (b : Nat) (h4 : a * b = n \u2227 a < n \u2227 b < n) from h3\n have h5 : 2 \u2264 a := by\n by_contra h6\n have h7 : a \u2264 1 := by linarith\n have h8 : n \u2264 b :=\n calc n\n _ = a * b := h4.left.symm\n _ \u2264 1 * b := by rel [h7]\n _ = b := by ring\n linarith --n \u2264 b contradicts b < n\n done\n have h6 : \u2203 (p : Nat), prime_factor p a := ih a h4.right.left h5\n obtain (p : Nat) (h7 : prime_factor p a) from h6\n apply Exists.intro p\n define --Goal : prime p \u2227 p \u2223 n\n define at h7 --h7 : prime p \u2227 p \u2223 a\n apply And.intro h7.left\n have h8 : a \u2223 n := by\n apply Exists.intro b\n show n = a * b from (h4.left).symm\n done\n show p \u2223 n from dvd_trans h7.right h8\n done\n done\n\nlemma exists_least_prime_factor {n : Nat} (h : 2 \u2264 n) :\n \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := by\n set S : Set Nat := {p : Nat | prime_factor p n}\n have h2 : \u2203 (p : Nat), p \u2208 S := exists_prime_factor n h\n show \u2203 (p : Nat), prime_factor p n \u2227\n \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q from well_ord_princ S h2\n done\n\nlemma all_prime_nil : all_prime [] := by\n define --Goal : \u2200 p \u2208 [], prime p\n fix p : Nat\n contrapos --Goal : \u00acprime p \u2192 p \u2209 []\n assume h1 : \u00acprime p\n show p \u2209 [] from List.not_mem_nil p\n done\n\nlemma all_prime_cons (n : Nat) (L : List Nat) :\n all_prime (n :: L) \u2194 prime n \u2227 all_prime L := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : all_prime (n :: L) --Goal : prime n \u2227 all_prime L\n define at h1 --h1 : \u2200 p \u2208 n :: L, prime p\n apply And.intro (h1 n (List.mem_cons_self n L))\n define --Goal : \u2200 p \u2208 L, prime p\n fix p : Nat\n assume h2 : p \u2208 L\n show prime p from h1 p (List.mem_cons_of_mem n h2)\n done\n \u00b7 -- (\u2190)\n assume h1 : prime n \u2227 all_prime L --Goal : all_prime (n :: l)\n define : all_prime L at h1\n define\n fix p : Nat\n assume h2 : p \u2208 n :: L\n rewrite [List.mem_cons] at h2 --h2 : p = n \u2228 p \u2208 L\n by_cases on h2\n \u00b7 -- Case 1. h2 : p = n\n rewrite [h2]\n show prime n from h1.left\n done\n \u00b7 -- Case 2. h2 : p \u2208 L\n show prime p from h1.right p h2\n done\n done\n done\n\nlemma nondec_nil : nondec [] := by\n define --Goal : True\n trivial --trivial proves some obviously true statements, such as True\n done\n\nlemma nondec_cons (n : Nat) (L : List Nat) :\n nondec (n :: L) \u2194 (\u2200 m \u2208 L, n \u2264 m) \u2227 nondec L := by rfl\n\nlemma prod_nil : prod [] = 1 := by rfl\n\nlemma prod_cons : prod (n :: L) = n * (prod L) := by rfl\n\nlemma exists_cons_of_length_eq_succ {A : Type}\n {l : List A} {n : Nat} (h : l.length = n + 1) :\n \u2203 (a : A) (L : List A), l = a :: L \u2227 L.length = n := by\n have h1 : \u00acl.length = 0 := by linarith\n rewrite [List.length_eq_zero] at h1\n obtain (a : A) (h2 : \u2203 (L : List A), l = a :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List A) (h3 : l = a :: L) from h2\n apply Exists.intro a\n apply Exists.intro L\n apply And.intro h3\n have h4 : (a :: L).length = L.length + 1 := List.length_cons a L\n rewrite [\u2190h3, h] at h4\n show L.length = n from (Nat.add_right_cancel h4).symm\n done\n\nlemma list_elt_dvd_prod_by_length (a : Nat) : \u2200 (n : Nat),\n \u2200 (l : List Nat), l.length = n \u2192 a \u2208 l \u2192 a \u2223 prod l := by\n by_induc\n \u00b7 --Base Case\n fix l : List Nat\n assume h1 : l.length = 0\n rewrite [List.length_eq_zero] at h1 --h1 : l = []\n rewrite [h1] --Goal : a \u2208 [] \u2192 a \u2223 prod []\n contrapos\n assume h2 : \u00aca \u2223 prod []\n show a \u2209 [] from List.not_mem_nil a\n done\n \u00b7 -- Induction Step\n fix n : Nat\n assume ih : \u2200 (l : List Nat), List.length l = n \u2192 a \u2208 l \u2192 a \u2223 prod l\n fix l : List Nat\n assume h1 : l.length = n + 1 --Goal : a \u2208 l \u2192 a \u2223 prod l\n obtain (b : Nat) (h2 : \u2203 (L : List Nat),\n l = b :: L \u2227 L.length = n) from exists_cons_of_length_eq_succ h1\n obtain (L : List Nat) (h3 : l = b :: L \u2227 L.length = n) from h2\n have h4 : a \u2208 L \u2192 a \u2223 prod L := ih L h3.right\n assume h5 : a \u2208 l\n rewrite [h3.left, prod_cons] --Goal : a \u2223 b * prod L\n rewrite [h3.left, List.mem_cons] at h5 --h5 : a = b \u2228 a \u2208 L\n by_cases on h5\n \u00b7 -- Case 1. h5 : a = b\n apply Exists.intro (prod L)\n rewrite [h5]\n rfl\n done\n \u00b7 -- Case 2. h5 : a \u2208 L\n have h6 : a \u2223 prod L := h4 h5\n have h7 : prod L \u2223 b * prod L := by\n apply Exists.intro b\n ring\n done\n show a \u2223 b * prod L from dvd_trans h6 h7\n done\n done\n done\n\nlemma list_elt_dvd_prod {a : Nat} {l : List Nat}\n (h : a \u2208 l) : a \u2223 prod l := by\n set n : Nat := l.length\n have h1 : l.length = n := by rfl\n show a \u2223 prod l from list_elt_dvd_prod_by_length a n l h1 h\n done\n\nlemma exists_prime_factorization : \u2200 (n : Nat), n \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n l := by\n by_strong_induc\n fix n : Nat\n assume ih : \u2200 n_1 < n, n_1 \u2265 1 \u2192\n \u2203 (l : List Nat), prime_factorization n_1 l\n assume h1 : n \u2265 1\n by_cases h2 : n = 1\n \u00b7 -- Case 1. h2 : n = 1\n apply Exists.intro []\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list []\n define\n show all_prime [] \u2227 nondec [] from\n And.intro all_prime_nil nondec_nil\n done\n \u00b7 -- Proof of prod [] = n\n rewrite [prod_nil, h2]\n rfl\n done\n done\n \u00b7 -- Case 2. h2 : n \u2260 1\n have h3 : n \u2265 2 := lt_of_le_of_ne' h1 h2\n obtain (p : Nat) (h4 : prime_factor p n \u2227 \u2200 (q : Nat),\n prime_factor q n \u2192 p \u2264 q) from exists_least_prime_factor h3\n have p_prime_factor : prime_factor p n := h4.left\n define at p_prime_factor\n have p_prime : prime p := p_prime_factor.left\n have p_dvd_n : p \u2223 n := p_prime_factor.right\n have p_least : \u2200 (q : Nat), prime_factor q n \u2192 p \u2264 q := h4.right\n obtain (m : Nat) (n_eq_pm : n = p * m) from p_dvd_n\n have h5 : m \u2260 0 := by\n contradict h1 with h6\n have h7 : n = 0 :=\n calc n\n _ = p * m := n_eq_pm\n _ = p * 0 := by rw [h6]\n _ = 0 := by ring\n rewrite [h7]\n decide\n done\n have m_pos : 0 < m := Nat.pos_of_ne_zero h5\n have m_lt_n : m < n := by\n define at p_prime\n show m < n from\n calc m\n _ < m + m := by linarith\n _ = 2 * m := by ring\n _ \u2264 p * m := by rel [p_prime.left]\n _ = n := n_eq_pm.symm\n done\n obtain (L : List Nat) (h6 : prime_factorization m L)\n from ih m m_lt_n m_pos\n define at h6\n have ndpl_L : nondec_prime_list L := h6.left\n define at ndpl_L\n apply Exists.intro (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of nondec_prime_list (p :: L)\n define\n apply And.intro\n \u00b7 -- Proof of all_prime (p :: L)\n rewrite [all_prime_cons]\n show prime p \u2227 all_prime L from And.intro p_prime ndpl_L.left\n done\n \u00b7 -- Proof of nondec (p :: L)\n rewrite [nondec_cons]\n apply And.intro _ ndpl_L.right\n fix q : Nat\n assume q_in_L : q \u2208 L\n have h7 : q \u2223 prod L := list_elt_dvd_prod q_in_L\n rewrite [h6.right] at h7 --h7 : q \u2223 m\n have h8 : m \u2223 n := by\n apply Exists.intro p\n rewrite [n_eq_pm]\n ring\n done\n have q_dvd_n : q \u2223 n := dvd_trans h7 h8\n have ap_L : all_prime L := ndpl_L.left\n define at ap_L\n have q_prime_factor : prime_factor q n :=\n And.intro (ap_L q q_in_L) q_dvd_n\n show p \u2264 q from p_least q q_prime_factor\n done\n done\n \u00b7 -- Proof of prod (p :: L) = n\n rewrite [prod_cons, h6.right, n_eq_pm]\n rfl\n done\n done\n done\n\ntheorem Theorem_7_2_2 {a b c : Nat}\n (h1 : c \u2223 a * b) (h2 : rel_prime a c) : c \u2223 b := by\n rewrite [\u2190Int.natCast_dvd_natCast] --Goal : \u2191c \u2223 \u2191b\n define at h1; define at h2; define\n obtain (j : Nat) (h3 : a * b = c * j) from h1\n set s : Int := gcd_c1 a c\n set t : Int := gcd_c2 a c\n have h4 : s * \u2191a + t * \u2191c = \u2191(gcd a c) := gcd_lin_comb c a\n rewrite [h2, Nat.cast_one] at h4 --h4 : s * \u2191a + t * \u2191c = (1 : Int)\n apply Exists.intro (s * \u2191j + t * \u2191b)\n show \u2191b = \u2191c * (s * \u2191j + t * \u2191b) from\n calc \u2191b\n _ = (1 : Int) * \u2191b := (one_mul _).symm\n _ = (s * \u2191a + t * \u2191c) * \u2191b := by rw [h4]\n _ = s * (\u2191a * \u2191b) + t * \u2191c * \u2191b := by ring\n _ = s * (\u2191c * \u2191j) + t * \u2191c * \u2191b := by\n rw [\u2190Nat.cast_mul a b, h3, Nat.cast_mul c j]\n _ = \u2191c * (s * \u2191j + t * \u2191b) := by ring\n done\n\nlemma le_nonzero_prod_left {a b : Nat} (h : a * b \u2260 0) : a \u2264 a * b := by\n have h1 : b \u2260 0 := by\n contradict h with h1\n rewrite [h1]\n ring\n done\n have h2 : 1 \u2264 b := Nat.pos_of_ne_zero h1\n show a \u2264 a * b from\n calc a\n = a * 1 := (mul_one a).symm\n _ \u2264 a * b := by rel [h2]\n done\n\nlemma le_nonzero_prod_right {a b : Nat} (h : a * b \u2260 0) : b \u2264 a * b := by\n rewrite [mul_comm]\n rewrite [mul_comm] at h\n show b \u2264 b * a from le_nonzero_prod_left h\n done\n\nlemma dvd_prime {a p : Nat}\n (h1 : prime p) (h2 : a \u2223 p) : a = 1 \u2228 a = p := sorry\n\nlemma rel_prime_of_prime_not_dvd {a p : Nat}\n (h1 : prime p) (h2 : \u00acp \u2223 a) : rel_prime a p := by\n have h3 : gcd a p \u2223 a := gcd_dvd_left a p\n have h4 : gcd a p \u2223 p := gcd_dvd_right a p\n have h5 : gcd a p = 1 \u2228 gcd a p = p := dvd_prime h1 h4\n have h6 : gcd a p \u2260 p := by\n contradict h2 with h6\n rewrite [h6] at h3\n show p \u2223 a from h3\n done\n disj_syll h5 h6\n show rel_prime a p from h5\n done\n\ntheorem Theorem_7_2_3 {a b p : Nat}\n (h1 : prime p) (h2 : p \u2223 a * b) : p \u2223 a \u2228 p \u2223 b := by\n or_right with h3\n have h4 : rel_prime a p := rel_prime_of_prime_not_dvd h1 h3\n show p \u2223 b from Theorem_7_2_2 h2 h4\n done\n\nlemma ge_one_of_prod_one {a b : Nat} (h : a * b = 1) : a \u2265 1 := by\n have h1 : a \u2260 0 := by\n by_contra h1\n rewrite [h1] at h\n contradict h\n linarith\n done\n show a \u2265 1 from Nat.pos_of_ne_zero h1\n done\n\nlemma eq_one_of_prod_one {a b : Nat} (h : a * b = 1) : a = 1 := by\n have h1 : a \u2265 1 := ge_one_of_prod_one h\n have h2 : a * b \u2260 0 := by linarith\n have h3 : a \u2264 a * b := le_nonzero_prod_left h2\n rewrite [h] at h3\n show a = 1 from Nat.le_antisymm h3 h1\n done\n\nlemma eq_one_of_dvd_one {n : Nat} (h : n \u2223 1) : n = 1 := by\n obtain (j : Nat) (h1 : 1 = n * j) from h\n show n = 1 from eq_one_of_prod_one h1.symm\n done\n\nlemma prime_not_one {p : Nat} (h : prime p) : p \u2260 1 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_4 {p : Nat} (h1 : prime p) :\n \u2200 (l : List Nat), p \u2223 prod l \u2192 \u2203 a \u2208 l, p \u2223 a := by\n apply List.rec\n \u00b7 -- Base Case. Goal : p \u2223 prod [] \u2192 \u2203 a \u2208 [], p \u2223 a\n rewrite [prod_nil]\n assume h2 : p \u2223 1\n show \u2203 a \u2208 [], p \u2223 a from\n absurd (eq_one_of_dvd_one h2) (prime_not_one h1)\n done\n \u00b7 -- Induction Step\n fix b : Nat\n fix L : List Nat\n assume ih : p \u2223 prod L \u2192 \u2203 a \u2208 L, p \u2223 a\n --Goal : p \u2223 prod (b :: L) \u2192 \u2203 a \u2208 b :: L, p \u2223 a\n assume h2 : p \u2223 prod (b :: L)\n rewrite [prod_cons] at h2\n have h3 : p \u2223 b \u2228 p \u2223 prod L := Theorem_7_2_3 h1 h2\n by_cases on h3\n \u00b7 -- Case 1. h3 : p \u2223 b\n apply Exists.intro b\n show b \u2208 b :: L \u2227 p \u2223 b from\n And.intro (List.mem_cons_self b L) h3\n done\n \u00b7 -- Case 2. h3 : p \u2223 prod L\n obtain (a : Nat) (h4 : a \u2208 L \u2227 p \u2223 a) from ih h3\n apply Exists.intro a\n show a \u2208 b :: L \u2227 p \u2223 a from\n And.intro (List.mem_cons_of_mem b h4.left) h4.right\n done\n done\n done\n\nlemma prime_in_list {p : Nat} {l : List Nat}\n (h1 : prime p) (h2 : all_prime l) (h3 : p \u2223 prod l) : p \u2208 l := by\n obtain (a : Nat) (h4 : a \u2208 l \u2227 p \u2223 a) from Theorem_7_2_4 h1 l h3\n define at h2\n have h5 : prime a := h2 a h4.left\n have h6 : p = 1 \u2228 p = a := dvd_prime h5 h4.right\n disj_syll h6 (prime_not_one h1)\n rewrite [h6]\n show a \u2208 l from h4.left\n done\n\nlemma first_le_first {p q : Nat} {l m : List Nat}\n (h1 : nondec_prime_list (p :: l)) (h2 : nondec_prime_list (q :: m))\n (h3 : prod (p :: l) = prod (q :: m)) : p \u2264 q := by\n define at h1; define at h2\n have h4 : q \u2223 prod (p :: l) := by\n define\n apply Exists.intro (prod m)\n rewrite [\u2190prod_cons]\n show prod (p :: l) = prod (q :: m) from h3\n done\n have h5 : all_prime (q :: m) := h2.left\n rewrite [all_prime_cons] at h5\n have h6 : q \u2208 p :: l := prime_in_list h5.left h1.left h4\n have h7 : nondec (p :: l) := h1.right\n rewrite [nondec_cons] at h7\n rewrite [List.mem_cons] at h6\n by_cases on h6\n \u00b7 -- Case 1. h6 : q = p\n linarith\n done\n \u00b7 -- Case 2. h6 : q \u2208 l\n have h8 : \u2200 m \u2208 l, p \u2264 m := h7.left\n show p \u2264 q from h8 q h6\n done\n done\n\nlemma nondec_prime_list_tail {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : nondec_prime_list l := by\n define at h\n define\n rewrite [all_prime_cons, nondec_cons] at h\n show all_prime l \u2227 nondec l from And.intro h.left.right h.right.right\n done\n\nlemma cons_prod_not_one {p : Nat} {l : List Nat}\n (h : nondec_prime_list (p :: l)) : prod (p :: l) \u2260 1 := by\n define at h\n have h1 : all_prime (p :: l) := h.left\n rewrite [all_prime_cons] at h1\n rewrite [prod_cons]\n by_contra h2\n show False from (prime_not_one h1.left) (eq_one_of_prod_one h2)\n done\n\nlemma list_nil_iff_prod_one {l : List Nat} (h : nondec_prime_list l) :\n l = [] \u2194 prod l = 1 := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : l = []\n rewrite [h1]\n show prod [] = 1 from prod_nil\n done\n \u00b7 -- (\u2190)\n contrapos\n assume h1 : \u00acl = []\n obtain (p : Nat) (h2 : \u2203 (L : List Nat), l = p :: L) from\n List.exists_cons_of_ne_nil h1\n obtain (L : List Nat) (h3 : l = p :: L) from h2\n rewrite [h3] at h\n rewrite [h3]\n show \u00acprod (p :: L) = 1 from cons_prod_not_one h\n done\n done\n\nlemma prime_pos {p : Nat} (h : prime p) : p > 0 := by\n define at h\n linarith\n done\n\ntheorem Theorem_7_2_5 : \u2200 (l1 l2 : List Nat),\n nondec_prime_list l1 \u2192 nondec_prime_list l2 \u2192\n prod l1 = prod l2 \u2192 l1 = l2 := by\n apply List.rec\n \u00b7 -- Base Case. Goal : \u2200 (l2 : List Nat), nondec_prime_list [] \u2192\n -- nondec_prime_list l2 \u2192 prod [] = prod l2 \u2192 [] = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list []\n assume h2 : nondec_prime_list l2\n assume h3 : prod [] = prod l2\n rewrite [prod_nil, eq_comm, \u2190list_nil_iff_prod_one h2] at h3\n show [] = l2 from h3.symm\n done\n \u00b7 -- Induction Step\n fix p : Nat\n fix L1 : List Nat\n assume ih : \u2200 (L2 : List Nat), nondec_prime_list L1 \u2192\n nondec_prime_list L2 \u2192 prod L1 = prod L2 \u2192 L1 = L2\n -- Goal : \u2200 (l2 : List Nat), nondec_prime_list (p :: L1) \u2192\n -- nondec_prime_list l2 \u2192 prod (p :: L1) = prod l2 \u2192 p :: L1 = l2\n fix l2 : List Nat\n assume h1 : nondec_prime_list (p :: L1)\n assume h2 : nondec_prime_list l2\n assume h3 : prod (p :: L1) = prod l2\n have h4 : \u00acprod (p :: L1) = 1 := cons_prod_not_one h1\n rewrite [h3, \u2190list_nil_iff_prod_one h2] at h4\n obtain (q : Nat) (h5 : \u2203 (L : List Nat), l2 = q :: L) from\n List.exists_cons_of_ne_nil h4\n obtain (L2 : List Nat) (h6 : l2 = q :: L2) from h5\n rewrite [h6] at h2 --h2 : nondec_prime_list (q :: L2)\n rewrite [h6] at h3 --h3 : prod (p :: L1) = prod (q :: L2)\n have h7 : p \u2264 q := first_le_first h1 h2 h3\n have h8 : q \u2264 p := first_le_first h2 h1 h3.symm\n have h9 : p = q := by linarith\n rewrite [h9, prod_cons, prod_cons] at h3\n --h3 : q * prod L1 = q * prod L2\n have h10 : nondec_prime_list L1 := nondec_prime_list_tail h1\n have h11 : nondec_prime_list L2 := nondec_prime_list_tail h2\n define at h2\n have h12 : all_prime (q :: L2) := h2.left\n rewrite [all_prime_cons] at h12\n have h13 : q > 0 := prime_pos h12.left\n have h14 : prod L1 = prod L2 := Nat.eq_of_mul_eq_mul_left h13 h3\n have h15 : L1 = L2 := ih L2 h10 h11 h14\n rewrite [h6, h9, h15]\n rfl\n done\n done\n\ntheorem fund_thm_arith (n : Nat) (h : n \u2265 1) :\n \u2203! (l : List Nat), prime_factorization n l := by\n exists_unique\n \u00b7 -- Existence\n show \u2203 (l : List Nat), prime_factorization n l from\n exists_prime_factorization n h\n done\n \u00b7 -- Uniqueness\n fix l1 : List Nat; fix l2 : List Nat\n assume h1 : prime_factorization n l1\n assume h2 : prime_factorization n l2\n define at h1; define at h2\n have h3 : prod l1 = n := h1.right\n rewrite [\u2190h2.right] at h3\n show l1 = l2 from Theorem_7_2_5 l1 l2 h1.left h2.left h3\n done\n done\n\n/- Section 7.3 -/\ntheorem congr_refl (m : Nat) : \u2200 (a : Int), a \u2261 a (MOD m) := by\n fix a : Int\n define --Goal : \u2203 (c : Int), a - a = \u2191m * c\n apply Exists.intro 0\n ring\n done\n\ntheorem congr_symm {m : Nat} : \u2200 {a b : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 a (MOD m) := by\n fix a : Int; fix b : Int\n assume h1 : a \u2261 b (MOD m)\n define at h1 --h1 : \u2203 (c : Int), a - b = \u2191m * c\n define --Goal : \u2203 (c : Int), b - a = \u2191m * c\n obtain (c : Int) (h2 : a - b = m * c) from h1\n apply Exists.intro (-c)\n show b - a = m * (-c) from\n calc b - a\n _ = -(a - b) := by ring\n _ = -(m * c) := by rw [h2]\n _ = m * (-c) := by ring\n done\n\ntheorem congr_trans {m : Nat} : \u2200 {a b c : Int},\n a \u2261 b (MOD m) \u2192 b \u2261 c (MOD m) \u2192 a \u2261 c (MOD m) := sorry\n\n/- Fundamental properties of congruence classes -/\nlemma cc_eq_iff_val_eq {n : Nat} (X Y : ZMod (n + 1)) :\n X = Y \u2194 X.val = Y.val := Fin.ext_iff\n\nlemma val_nat_eq_mod (n k : Nat) :\n ([k]_(n + 1)).val = k % (n + 1) := by rfl\n\nlemma val_zero (n : Nat) : ([0]_(n + 1)).val = 0 := by rfl\n\ntheorem cc_rep {m : Nat} (X : ZMod m) : \u2203 (a : Int), X = [a]_m :=\n match m with\n | 0 => by\n apply Exists.intro X\n rfl\n done\n | n + 1 => by\n apply Exists.intro \u2191(X.val)\n have h1 : X.val < n + 1 := Fin.prop X\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, Nat.mod_eq_of_lt h1]\n rfl\n done\n\ntheorem add_class (m : Nat) (a b : Int) :\n [a]_m + [b]_m = [a + b]_m := (Int.cast_add a b).symm\n\ntheorem mul_class (m : Nat) (a b : Int) :\n [a]_m * [b]_m = [a * b]_m := (Int.cast_mul a b).symm\n\nlemma cc_eq_iff_sub_zero (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 [a - b]_m = [0]_m := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : [a]_m = [b]_m\n have h2 : a - b = a + (-b) := by ring\n have h3 : b + (-b) = 0 := by ring\n show [a - b]_m = [0]_m from\n calc [a - b]_m\n _ = [a + (-b)]_m := by rw [h2]\n _ = [a]_m + [-b]_m := by rw [add_class]\n _ = [b]_m + [-b]_m := by rw [h1]\n _ = [b + -b]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n \u00b7 -- (\u2190)\n assume h1 : [a - b]_m = [0]_m\n have h2 : b + (a - b) = a := by ring\n have h3 : b + 0 = b := by ring\n show [a]_m = [b]_m from\n calc [a]_m\n _ = [b + (a - b)]_m := by rw [h2]\n _ = [b]_m + [a - b]_m := by rw [add_class]\n _ = [b]_m + [0]_m := by rw [h1]\n _ = [b + 0]_m := by rw [add_class]\n _ = [b]_m := by rw [h3]\n done\n done\n\nlemma cc_neg_zero_of_cc_zero (m : Nat) (a : Int) :\n [a]_m = [0]_m \u2192 [-a]_m = [0]_m := by\n assume h1 : [a]_m = [0]_m\n have h2 : 0 + (-a) = -a := by ring\n have h3 : a + (-a) = 0 := by ring\n show [-a]_m = [0]_m from\n calc [-a]_m\n _ = [0 + (-a)]_m := by rw [h2]\n _ = [0]_m + [-a]_m := by rw [add_class]\n _ = [a]_m + [-a]_m := by rw [h1]\n _ = [a + (-a)]_m := by rw [add_class]\n _ = [0]_m := by rw [h3]\n done\n\nlemma cc_neg_zero_iff_cc_zero (m : Nat) (a : Int) :\n [-a]_m = [0]_m \u2194 [a]_m = [0]_m := by\n apply Iff.intro _ (cc_neg_zero_of_cc_zero m a)\n assume h1 : [-a]_m = [0]_m\n have h2 : [-(-a)]_m = [0]_m := cc_neg_zero_of_cc_zero m (-a) h1\n have h3 : -(-a) = a := by ring\n rewrite [h3] at h2\n show [a]_m = [0]_m from h2\n done\n\nlemma cc_mod_0 (a : Int) : [a]_0 = a := by rfl\n\nlemma cc_nat_zero_iff_dvd (m k : Nat) : [k]_m = [0]_m \u2194 m \u2223 k :=\n match m with\n | 0 => by\n have h : (0 : Int) = (\u2191(0 : Nat) : Int) := by rfl\n rewrite [cc_mod_0, cc_mod_0, h, Nat.cast_inj]\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : k = 0\n rewrite [h1]\n show 0 \u2223 0 from dvd_self 0\n done\n \u00b7 -- (\u2190)\n assume h1 : 0 \u2223 k\n obtain (c : Nat) (h2 : k = 0 * c) from h1\n rewrite [h2]\n ring\n done\n done\n | n + 1 => by\n rewrite [cc_eq_iff_val_eq, val_nat_eq_mod, val_zero]\n show k % (n + 1) = 0 \u2194 n + 1 \u2223 k from\n (Nat.dvd_iff_mod_eq_zero (n + 1) k).symm\n done\n\nlemma cc_zero_iff_dvd (m : Nat) (a : Int) : [a]_m = [0]_m \u2194 \u2191m \u2223 a := by\n obtain (k : Nat) (h1 : a = \u2191k \u2228 a = -\u2191k) from Int.eq_nat_or_neg a\n by_cases on h1\n \u00b7 -- Case 1. h1: a = \u2191k\n rewrite [h1, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n \u00b7 -- Case 2. h1: a = -\u2191k\n rewrite [h1, cc_neg_zero_iff_cc_zero, Int.dvd_neg, Int.natCast_dvd_natCast]\n show [\u2191k]_m = [0]_m \u2194 m \u2223 k from cc_nat_zero_iff_dvd m k\n done\n done\n\ntheorem cc_eq_iff_congr (m : Nat) (a b : Int) :\n [a]_m = [b]_m \u2194 a \u2261 b (MOD m) :=\n calc [a]_m = [b]_m\n _ \u2194 [a - b]_m = [0]_m := cc_eq_iff_sub_zero m a b\n _ \u2194 \u2191m \u2223 (a - b) := cc_zero_iff_dvd m (a - b)\n _ \u2194 a \u2261 b (MOD m) := by rfl\n/- End of fundamental properties of congruence classes -/\n\nlemma mod_nonneg (m : Nat) [NeZero m] (a : Int) : 0 \u2264 a % m := by\n have h1 : (\u2191m : Int) \u2260 0 := (Nat.cast_ne_zero).rtl (NeZero.ne m)\n show 0 \u2264 a % m from Int.emod_nonneg a h1\n done\n\nlemma mod_lt (m : Nat) [NeZero m] (a : Int) : a % m < m := by\n have h1 : m > 0 := Nat.pos_of_ne_zero (NeZero.ne m)\n have h2 : (\u2191m : Int) > 0 := (Nat.cast_pos).rtl h1\n show a % m < m from Int.emod_lt_of_pos a h2\n done\n\nlemma congr_mod_mod (m : Nat) (a : Int) : a \u2261 a % m (MOD m) := by\n define\n have h1 : m * (a / m) + a % m = a := Int.ediv_add_emod a m\n apply Exists.intro (a / m)\n show a - a % m = m * (a / m) from\n calc a - (a % m)\n _ = m * (a / m) + a % m - a % m := by rw [h1]\n _ = m * (a / m) := by ring\n done\n\nlemma mod_cmpl_res (m : Nat) [NeZero m] (a : Int) :\n 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) :=\n And.intro (mod_nonneg m a) (And.intro (mod_lt m a) (congr_mod_mod m a))\n\ntheorem Theorem_7_3_1 (m : Nat) [NeZero m] (a : Int) :\n \u2203! (r : Int), 0 \u2264 r \u2227 r < m \u2227 a \u2261 r (MOD m) := by\n exists_unique\n \u00b7 -- Existence\n apply Exists.intro (a % m)\n show 0 \u2264 a % m \u2227 a % m < m \u2227 a \u2261 a % m (MOD m) from\n mod_cmpl_res m a\n done\n \u00b7 -- Uniqueness\n fix r1 : Int; fix r2 : Int\n assume h1 : 0 \u2264 r1 \u2227 r1 < m \u2227 a \u2261 r1 (MOD m)\n assume h2 : 0 \u2264 r2 \u2227 r2 < m \u2227 a \u2261 r2 (MOD m)\n have h3 : r1 \u2261 r2 (MOD m) :=\n congr_trans (congr_symm h1.right.right) h2.right.right\n obtain (d : Int) (h4 : r1 - r2 = m * d) from h3\n have h5 : r1 - r2 < m * 1 := by linarith\n have h6 : m * (-1) < r1 - r2 := by linarith\n rewrite [h4] at h5 --h5 : m * d < m * 1\n rewrite [h4] at h6 --h6 : m * -1 < m * d\n have h7 : (\u2191m : Int) \u2265 0 := Nat.cast_nonneg m\n have h8 : d < 1 := lt_of_mul_lt_mul_of_nonneg_left h5 h7\n have h9 : -1 < d := lt_of_mul_lt_mul_of_nonneg_left h6 h7\n have h10 : d = 0 := by linarith\n show r1 = r2 from\n calc r1\n _ = r1 - r2 + r2 := by ring\n _ = m * 0 + r2 := by rw [h4, h10]\n _ = r2 := by ring\n done\n done\n\nlemma cc_eq_mod (m : Nat) (a : Int) : [a]_m = [a % m]_m :=\n (cc_eq_iff_congr m a (a % m)).rtl (congr_mod_mod m a)\n\ntheorem Theorem_7_3_6_1 {m : Nat} (X Y : ZMod m) : X + Y = Y + X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n obtain (b : Int) (h2 : Y = [b]_m) from cc_rep Y\n rewrite [h1, h2]\n have h3 : a + b = b + a := by ring\n show [a]_m + [b]_m = [b]_m + [a]_m from\n calc [a]_m + [b]_m\n _ = [a + b]_m := add_class m a b\n _ = [b + a]_m := by rw [h3]\n _ = [b]_m + [a]_m := (add_class m b a).symm\n done\n\ntheorem Theorem_7_3_6_7 {m : Nat} (X : ZMod m) : X * [1]_m = X := by\n obtain (a : Int) (h1 : X = [a]_m) from cc_rep X\n rewrite [h1]\n have h2 : a * 1 = a := by ring\n show [a]_m * [1]_m = [a]_m from\n calc [a]_m * [1]_m\n _ = [a * 1]_m := mul_class m a 1\n _ = [a]_m := by rw [h2]\n done\n\ntheorem Exercise_7_2_6 (a b : Nat) :\n rel_prime a b \u2194 \u2203 (s t : Int), s * a + t * b = 1 := sorry\n\nlemma gcd_c2_inv {m a : Nat} (h1 : rel_prime m a) :\n [a]_m * [gcd_c2 m a]_m = [1]_m := by\n set s : Int := gcd_c1 m a\n have h2 : s * m + (gcd_c2 m a) * a = gcd m a := gcd_lin_comb a m\n define at h1\n rewrite [h1, Nat.cast_one] at h2 --h2 : s * \u2191m + gcd_c2 m a * \u2191a = 1\n rewrite [mul_class, cc_eq_iff_congr]\n define --Goal : \u2203 (c : Int), \u2191a * gcd_c2 m a - 1 = \u2191m * c\n apply Exists.intro (-s)\n show a * (gcd_c2 m a) - 1 = m * (-s) from\n calc a * (gcd_c2 m a) - 1\n _ = s * m + (gcd_c2 m a) * a + m * (-s) - 1 := by ring\n _ = 1 + m * (-s) - 1 := by rw [h2]\n _ = m * (-s) := by ring\n done\n\ntheorem Theorem_7_3_7 (m a : Nat) :\n invertible [a]_m \u2194 rel_prime m a := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h1 : invertible [a]_m\n define at h1\n obtain (Y : ZMod m) (h2 : [a]_m * Y = [1]_m) from h1\n obtain (b : Int) (h3 : Y = [b]_m) from cc_rep Y\n rewrite [h3, mul_class, cc_eq_iff_congr] at h2\n define at h2\n obtain (c : Int) (h4 : a * b - 1 = m * c) from h2\n rewrite [Exercise_7_2_6]\n --Goal : \u2203 (s t : Int), s * \u2191m + t * \u2191a = 1\n apply Exists.intro (-c)\n apply Exists.intro b\n show (-c) * m + b * a = 1 from\n calc (-c) * m + b * a\n _ = (-c) * m + (a * b - 1) + 1 := by ring\n _ = (-c) * m + m * c + 1 := by rw [h4]\n _ = 1 := by ring\n done\n \u00b7 -- (\u2190)\n assume h1 : rel_prime m a\n define\n show \u2203 (Y : ZMod m), [a]_m * Y = [1]_m from\n Exists.intro [gcd_c2 m a]_m (gcd_c2_inv h1)\n done\n done\n\n/- Section 7.4 -/\nsection Euler\nopen Euler\n\nlemma num_rp_below_base {m : Nat} :\n num_rp_below m 0 = 0 := by rfl\n\nlemma num_rp_below_step_rp {m j : Nat} (h : rel_prime m j) :\n num_rp_below m (j + 1) = (num_rp_below m j) + 1 := by\n have h1 : num_rp_below m (j + 1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : gcd m j = 1\n rewrite [if_pos h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j + 1\n show num_rp_below m (j + 1) = num_rp_below m j + 1 from h1\n done\n\nlemma num_rp_below_step_not_rp {m j : Nat} (h : \u00acrel_prime m j) :\n num_rp_below m (j + 1) = num_rp_below m j := by\n have h1 : num_rp_below m (j +1) =\n if gcd m j = 1 then (num_rp_below m j) + 1\n else num_rp_below m j := by rfl\n define at h --h : \u00acgcd m j = 1\n rewrite [if_neg h] at h1\n --h1 : num_rp_below m (j + 1) = num_rp_below m j\n show num_rp_below m (j + 1) = num_rp_below m j from h1\n done\n\nlemma phi_def (m : Nat) : phi m = num_rp_below m m := by rfl\n\n#eval phi 10 --Answer: 4\n\nlemma prod_inv_iff_inv {m : Nat} {X : ZMod m}\n (h1 : invertible X) (Y : ZMod m) :\n invertible (X * Y) \u2194 invertible Y := by\n apply Iff.intro\n \u00b7 -- (\u2192)\n assume h2 : invertible (X * Y)\n obtain (Z : ZMod m) (h3 : X * Y * Z = [1]_m) from h2\n apply Exists.intro (X * Z)\n rewrite [\u2190h3] --Goal : Y * (X * Z) = X * Y * Z\n ring --Note that ring can do algebra in ZMod m\n done\n \u00b7 -- (\u2190)\n assume h2 : invertible Y\n obtain (Xi : ZMod m) (h3 : X * Xi = [1]_m) from h1\n obtain (Yi : ZMod m) (h4 : Y * Yi = [1]_m) from h2\n apply Exists.intro (Xi * Yi)\n show (X * Y) * (Xi * Yi) = [1]_m from\n calc X * Y * (Xi * Yi)\n _ = (X * Xi) * (Y * Yi) := by ring\n _ = [1]_m * [1]_m := by rw [h3, h4]\n _ = [1]_m := Theorem_7_3_6_7 [1]_m\n done\n done\n\nlemma F_rp_def {m i : Nat} (h : rel_prime m i) :\n F m i = [i]_m := by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h --h : gcd m i = 1\n rewrite [if_pos h] at h1\n show F m i = [i]_m from h1\n done\n\n", "theoremStatement": "lemma F_not_rp_def {m i : Nat} (h : \u00acrel_prime m i) :\n F m i = [1]_m ", "theoremName": "HTPI.F_not_rp_def", "fileCreated": {"commit": "c83e1d1287455029a0dba8686f9d2aca166d412b", "date": "2023-04-04"}, "theoremCreated": {"commit": "515de92e6c12bba8178aad612650f80eacd0a245", "date": "2023-04-13"}, "file": "htpi/HTPILib/Chap7.lean", "module": "HTPILib.Chap7", "jsonFile": "HTPILib.Chap7.jsonl", 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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.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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", "Mathlib.Tactic.Rify", "Mathlib.Tactic.Sat.FromLRAT", "Mathlib.Control.Monad.Writer", "Mathlib.Init.Control.Lawful", "Mathlib.Control.Monad.Cont", "Mathlib.Control.ULiftable", "Mathlib.Control.Random", "Mathlib.Testing.SlimCheck.Gen", "Mathlib.Testing.SlimCheck.Sampleable", "Mathlib.Testing.SlimCheck.Testable", "Mathlib.Tactic.SlimCheck", "ProofWidgets.Component.PenroseDiagram", "ProofWidgets.Presentation.Expr", "Mathlib.Tactic.Widget.CommDiag", "Mathlib.Tactic.Widget.Gcongr", "Mathlib.Tactic", "Mathlib.Data.Rel", "HTPILib.HTPIDefs", "HTPILib.IntroLean", "HTPILib.Chap3", "HTPILib.Chap4", "HTPILib.Chap5", "HTPILib.Chap8Part1", "HTPILib.Chap6"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h1 : F m i = if gcd m i = 1 then [i]_m else [1]_m := by rfl\n define at h\n rewrite [h1, if_neg h]\n rfl\n done", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 124}} +{"srcContext": "/- Copyright 2023 Daniel J. Velleman -/\n\nimport Lean.Elab.Tactic\nimport Mathlib.Tactic\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Data.Set.Lattice\nimport Mathlib.Data.Rel\nimport Mathlib.Data.Nat.Defs\nimport Mathlib.Data.Int.Defs\nimport Mathlib.Data.Rat.Defs\nimport Mathlib.Data.Real.Basic\nimport Mathlib.Data.ZMod.Defs\n\ndef Iff.ltr {p q : Prop} (h : p \u2194 q) := h.mp\ndef Iff.rtl {p q : Prop} (h : p \u2194 q) := h.mpr\n\n--Make sure Lean understands {x} and \u2205 as Sets, not Finsets\nattribute [default_instance] Set.instSingletonSet\nattribute [default_instance] Set.instEmptyCollectionSet\n\n-- Used in one exercise in Chapter 3.\nnotation:50 a:50 \" \u2288 \" b:50 => \u00ac (a \u2286 b)\n\n--Note: Mathlib.Order.SymmDiff.lean defines this, but it is scoped[symmDiff] there.\n-- Use in one exercise in Chapter 3.\ninfixl:100 \" \u2206 \" => symmDiff\n\nnamespace HTPI\n--Some theorems not in library\n", "theoremStatement": "theorem not_not_and_distrib {p q : Prop} : \u00ac(\u00ac p \u2227 q) \u2194 (p \u2228 \u00ac q) ", "theoremName": "HTPI.not_not_and_distrib", "fileCreated": {"commit": "94654f92e3e5e5cd0ba367d89e9ba3f1ed7c911e", "date": "2022-09-27"}, "theoremCreated": {"commit": "cc48e6ba4c656bf74f7690323b2c27f2c466665b", "date": "2023-03-11"}, "file": "htpi/HTPILib/HTPIDefs.lean", "module": "HTPILib.HTPIDefs", "jsonFile": "HTPILib.HTPIDefs.jsonl", "positionMetadata": {"lineInFile": 30, "tokenPositionInFile": 859, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 12, "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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"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.SizeOf", "Lean.Meta.Injective", "Lean.Meta.IndPredBelow", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "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.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.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.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.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.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "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.Nontrivial.Defs", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Data.Sigma.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.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "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.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Data.Subtype", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "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.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Data.Int.Cast.Basic", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Mathlib.Tactic.NormNum.Result", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Init.Order.LinearOrder", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Use", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "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.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.IrreducibleDef", "Mathlib.Tactic.Lift", "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.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.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "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.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "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.Data.Nat.Cast.Commute", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Init.Propext", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Data.Int.Basic", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "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.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.Field.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Data.Nat.Order.Basic", "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.Rat.Defs", "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.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "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.Tactic.NormNum.OfScientific", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "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.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.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "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.Tactic.ApplyFun", "Mathlib.Tactic.ArithMult.Init", "Mathlib.Tactic.ArithMult", "Mathlib.Init.Data.Nat.Bitwise", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.CancelDenoms", "Mathlib.CategoryTheory.Category.Init", "Mathlib.Data.Opposite", "Mathlib.Combinatorics.Quiver.Basic", "Mathlib.CategoryTheory.Category.Basic", "Mathlib.CategoryTheory.Functor.Basic", "Mathlib.Util.AddRelatedDecl", "Mathlib.Tactic.CategoryTheory.Reassoc", "Mathlib.CategoryTheory.NatTrans", "Mathlib.CategoryTheory.Iso", "Mathlib.CategoryTheory.Functor.Category", "Mathlib.CategoryTheory.NatIso", "Mathlib.CategoryTheory.Bicategory.Basic", "Mathlib.CategoryTheory.Bicategory.Functor", "Mathlib.CategoryTheory.Bicategory.Free", "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "Mathlib.CategoryTheory.Functor.FullyFaithful", "Mathlib.CategoryTheory.FullSubcategory", "Mathlib.CategoryTheory.Whiskering", "Mathlib.CategoryTheory.EssentialImage", "Mathlib.Tactic.CategoryTheory.Slice", "Mathlib.CategoryTheory.Equivalence", "Mathlib.CategoryTheory.Opposites", "Mathlib.CategoryTheory.EqToHom", "Mathlib.CategoryTheory.Functor.Const", "Mathlib.CategoryTheory.Products.Basic", "Mathlib.CategoryTheory.Monoidal.Category", "Mathlib.CategoryTheory.Adjunction.Basic", "Mathlib.CategoryTheory.Monoidal.Functor", "Mathlib.CategoryTheory.Monoidal.Free.Basic", "Mathlib.Tactic.CategoryTheory.MonoidalComp", "Mathlib.Tactic.CategoryTheory.Coherence", "Mathlib.CategoryTheory.Pi.Basic", "Mathlib.Logic.Lemmas", "Mathlib.Combinatorics.Quiver.Path", "Mathlib.Combinatorics.Quiver.Push", "Mathlib.Combinatorics.Quiver.Symmetric", "Mathlib.CategoryTheory.Groupoid", "Mathlib.CategoryTheory.EpiMono", "Mathlib.CategoryTheory.Types", "Mathlib.CategoryTheory.Balanced", "Mathlib.CategoryTheory.Comma.Basic", "Mathlib.CategoryTheory.Comma.Arrow", "Mathlib.CategoryTheory.CommSq", "Mathlib.CategoryTheory.LiftingProperties.Basic", "Mathlib.CategoryTheory.Limits.Shapes.StrongEpi", "Mathlib.CategoryTheory.LiftingProperties.Adjunction", "Mathlib.CategoryTheory.Functor.EpiMono", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.CategoryTheory.DiscreteCategory", "Mathlib.CategoryTheory.Functor.Hom", "Mathlib.CategoryTheory.Functor.Currying", "Mathlib.CategoryTheory.Yoneda", "Mathlib.CategoryTheory.Functor.ReflectsIso", "Mathlib.CategoryTheory.Limits.Cones", "Mathlib.CategoryTheory.Limits.IsLimit", "Mathlib.CategoryTheory.Category.ULift", "Mathlib.CategoryTheory.Category.Preorder", "Mathlib.CategoryTheory.ConcreteCategory.Bundled", "Mathlib.CategoryTheory.Bicategory.Strict", "Mathlib.CategoryTheory.Category.Cat", "Mathlib.CategoryTheory.IsomorphismClasses", "Mathlib.CategoryTheory.Thin", "Mathlib.CategoryTheory.Skeletal", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.UnivLE", "Mathlib.Logic.Small.Basic", "Mathlib.CategoryTheory.EssentiallySmall", "Mathlib.CategoryTheory.Limits.HasLimits", "Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks", "Mathlib.CategoryTheory.PUnit", "Mathlib.CategoryTheory.PEmpty", "Mathlib.CategoryTheory.Limits.Shapes.Terminal", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Logic.Small.Set", "Mathlib.CategoryTheory.Comma.StructuredArrow", "Mathlib.CategoryTheory.Comma.Over", "Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts", "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Preserves.Basic", "Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks", "Mathlib.CategoryTheory.Limits.Constructions.EpiMono", "Mathlib.CategoryTheory.ConcreteCategory.Basic", "Mathlib.Tactic.CategoryTheory.Elementwise", "Mathlib.Tactic.Change", "Mathlib.Tactic.Clean", "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.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.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.Init.Data.Int.Bitwise", "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.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Field.IsField", "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.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.Set.List", "Mathlib.Data.List.Dedup", "Mathlib.Algebra.BigOperators.List.Defs", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Count", "Mathlib.Data.List.Perm", "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.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.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Order", "Mathlib.Data.List.MinMax", "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.Finset.Piecewise", "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.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "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.BigOperators.Order", "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.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Data.DFinsupp.Basic", "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.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.LinearAlgebra.Basic", "Mathlib.LinearAlgebra.Pi", "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.Algebra.MonoidAlgebra.Basic", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "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.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "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.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.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.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Polynomial.Basic", "Mathlib.Data.Polynomial.Monomial", "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.Polynomial.Coeff", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Data.Polynomial.Degree.Definitions", "Mathlib.Tactic.FinCases", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Data.Polynomial.Induction", "Mathlib.Data.Polynomial.Eval", "Mathlib.Data.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.ProxyType", "Mathlib.Tactic.DeriveFintype", "Mathlib.Control.Traversable.Lemmas", "Mathlib.Tactic.DeriveTraversable", "Mathlib.Tactic.Eval", "Mathlib.Tactic.Explode.Datatypes", "Mathlib.Tactic.Explode.Pretty", "Mathlib.Tactic.Explode", "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.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "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.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "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.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "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.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.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "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.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "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.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.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "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.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Init.Data.Subtype.Basic", "Mathlib.Algebra.Star.SelfAdjoint", "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.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.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.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.Analysis.Convex.Topology", "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.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "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.RingTheory.Localization.FractionRing", "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.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Data.MvPolynomial.Basic", "Mathlib.Data.MvPolynomial.Rename", "Mathlib.Data.MvPolynomial.Degrees", "Mathlib.Data.MvPolynomial.Variables", "Mathlib.Data.MvPolynomial.CommRing", "Mathlib.Data.Polynomial.AlgebraMap", "Mathlib.Data.MvPolynomial.Equiv", "Mathlib.Data.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Data.Polynomial.EraseLead", "Mathlib.Data.Polynomial.Inductions", "Mathlib.Data.Polynomial.Degree.TrailingDegree", "Mathlib.Data.Polynomial.Reverse", "Mathlib.Data.Polynomial.Monic", "Mathlib.Data.Polynomial.Div", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Data.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Data.Polynomial.FieldDivision", "Mathlib.Data.Polynomial.CancelLeads", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Data.Polynomial.Expand", "Mathlib.Data.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.FieldTheory.Finiteness", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.Data.Matrix.DMatrix", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.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.RingTheory.Adjoin.Tower", "Mathlib.Data.Polynomial.Module.Basic", "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.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap", "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.Asymptotics.Theta", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "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.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.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "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.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Tactic.FunProp.Measurable", "Mathlib.Tactic.FunProp.AEMeasurable", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "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.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "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.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.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Calculus.FDeriv.Pi", "Mathlib.Analysis.Calculus.Deriv.Inv", "Mathlib.Tactic.FunProp.Differentiable", "Mathlib.Tactic.FunProp.ContDiff", "Mathlib.Tactic.Generalize", "Mathlib.Tactic.Have", "Mathlib.Tactic.LiftLets", "Mathlib.Tactic.ModCases", "Mathlib.Tactic.MoveAdd", "Mathlib.Tactic.NoncommRing", "Mathlib.Tactic.NormNum.BigOperators", "Mathlib.Tactic.NormNum.IsCoprime", "Mathlib.Data.Fintype.Parity", "Mathlib.Data.Int.Range", "Mathlib.NumberTheory.LegendreSymbol.MulCharacter", "Mathlib.NumberTheory.LegendreSymbol.ZModChar", "Mathlib.Algebra.Squarefree.Basic", "Mathlib.Data.Polynomial.Lifts", "Mathlib.Data.Polynomial.Splits", "Mathlib.RingTheory.PowerBasis", "Mathlib.FieldTheory.Separable", "Mathlib.FieldTheory.Finite.Basic", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic", "Mathlib.NumberTheory.LegendreSymbol.Basic", "Mathlib.Data.PNat.Prime", "Mathlib.Algebra.IsPrimePow", "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.RingTheory.RootsOfUnity.Complex", "Mathlib.RingTheory.HahnSeries.Basic", "Mathlib.RingTheory.HahnSeries.Addition", "Mathlib.Data.Set.MulAntidiagonal", "Mathlib.Data.Finset.MulAntidiagonal", "Mathlib.RingTheory.HahnSeries.Multiplication", "Mathlib.Data.Finset.PiAntidiagonal", "Mathlib.RingTheory.MvPowerSeries.Basic", "Mathlib.RingTheory.PowerSeries.Basic", "Mathlib.Data.Finsupp.PWO", "Mathlib.RingTheory.HahnSeries.PowerSeries", "Mathlib.Algebra.Order.Group.WithTop", "Mathlib.RingTheory.Valuation.Basic", "Mathlib.RingTheory.HahnSeries.Summable", "Mathlib.RingTheory.LaurentSeries", "Mathlib.FieldTheory.RatFunc", "Mathlib.Data.Nat.Factorization.PrimePow", "Mathlib.Data.Nat.Squarefree", "Mathlib.NumberTheory.ArithmeticFunction", "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "Mathlib.RingTheory.MvPolynomial.Tower", "Mathlib.RingTheory.FinitePresentation", "Mathlib.RingTheory.QuotientNoetherian", "Mathlib.RingTheory.AdjoinRoot", "Mathlib.FieldTheory.IntermediateField", "Mathlib.RingTheory.Adjoin.Field", "Mathlib.FieldTheory.SplittingField.IsSplittingField", "Mathlib.FieldTheory.SplittingField.Construction", "Mathlib.RingTheory.Localization.Integer", "Mathlib.RingTheory.Localization.Integral", "Mathlib.RingTheory.IntegrallyClosed", "Mathlib.RingTheory.Polynomial.GaussLemma", "Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed", "Mathlib.RingTheory.Prime", "Mathlib.RingTheory.EisensteinCriterion", "Mathlib.RingTheory.Polynomial.Eisenstein.Basic", "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "Mathlib.RingTheory.RootsOfUnity.Minpoly", "Mathlib.RingTheory.Polynomial.Cyclotomic.Roots", "Mathlib.LinearAlgebra.FreeModule.PID", "Mathlib.LinearAlgebra.BilinearForm.Basic", "Mathlib.LinearAlgebra.BilinearForm.Hom", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.LinearAlgebra.BilinearForm.Properties", "Mathlib.LinearAlgebra.BilinearForm.DualLattice", "Mathlib.RingTheory.Localization.Ideal", "Mathlib.RingTheory.Localization.AtPrime", "Mathlib.RingTheory.Ideal.Over", "Mathlib.RingTheory.Localization.NumDen", "Mathlib.RingTheory.Polynomial.RationalRoot", "Mathlib.RingTheory.DedekindDomain.Basic", "Mathlib.Algebra.Module.LocalizedModule", "Mathlib.RingTheory.Localization.Module", "Mathlib.LinearAlgebra.Matrix.Nondegenerate", "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv", "Mathlib.LinearAlgebra.Basis.Bilinear", "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "Mathlib.LinearAlgebra.Matrix.BilinearForm", "Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly", "Mathlib.LinearAlgebra.Matrix.Block", "Mathlib.LinearAlgebra.Vandermonde", "Mathlib.LinearAlgebra.Contraction", "Mathlib.LinearAlgebra.Trace", "Mathlib.GroupTheory.Abelianization", "Mathlib.GroupTheory.FreeGroup.Basic", "Mathlib.GroupTheory.FreeAbelianGroup", "Mathlib.RingTheory.FreeRing", "Mathlib.RingTheory.FreeCommRing", "Mathlib.Algebra.DirectLimit", "Mathlib.FieldTheory.Adjoin", "Mathlib.FieldTheory.Extension", "Mathlib.GroupTheory.Perm.ViaEmbedding", "Mathlib.GroupTheory.Solvable", "Mathlib.FieldTheory.Normal", "Mathlib.FieldTheory.Perfect", "Mathlib.FieldTheory.IsAlgClosed.Basic", "Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure", "Mathlib.FieldTheory.PrimitiveElement", "Mathlib.Algebra.GroupRingAction.Invariant", "Mathlib.Algebra.Polynomial.GroupRingAction", "Mathlib.FieldTheory.Fixed", "Mathlib.FieldTheory.NormalClosure", "Mathlib.GroupTheory.GroupAction.FixedPoints", "Mathlib.GroupTheory.GroupAction.FixingSubgroup", "Mathlib.FieldTheory.Galois", "Mathlib.FieldTheory.Minpoly.MinpolyDiv", "Mathlib.RingTheory.Trace", "Mathlib.RingTheory.DedekindDomain.IntegralClosure", "Mathlib.NumberTheory.NumberField.Basic", "Mathlib.NumberTheory.Cyclotomic.Basic", "Mathlib.RingTheory.Adjoin.PowerBasis", "Mathlib.RingTheory.MvPolynomial.Symmetric", "Mathlib.RingTheory.Polynomial.Vieta", "Mathlib.Topology.Algebra.Polynomial", "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.Complex.Arg", "Mathlib.RingTheory.Polynomial.Cyclotomic.Eval", "Mathlib.RingTheory.Norm", "Mathlib.Data.ZMod.Algebra", "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "Mathlib.FieldTheory.Finite.GaloisField", "Mathlib.FieldTheory.Finite.Trace", "Mathlib.Algebra.Group.AddChar", "Mathlib.NumberTheory.LegendreSymbol.AddCharacter", "Mathlib.Algebra.CharP.CharAndCard", "Mathlib.NumberTheory.LegendreSymbol.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum", "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "Mathlib.Tactic.NormNum.LegendreSymbol", "Mathlib.Data.Nat.Fib.Basic", "Mathlib.Tactic.NormNum.NatFib", "Mathlib.Tactic.NormNum.NatSqrt", "Mathlib.Tactic.NormNum.Prime", "Mathlib.Tactic.Polyrith", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Tactic.ProdAssoc", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Recall", "Mathlib.Tactic.ReduceModChar.Ext", "Mathlib.Tactic.ReduceModChar", "Mathlib.Tactic.Replace", "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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