diff --git "a/minictx-valid/pfr_cross.jsonl" "b/minictx-valid/pfr_cross.jsonl" new file mode 100644--- /dev/null +++ "b/minictx-valid/pfr_cross.jsonl" @@ -0,0 +1,43 @@ +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\nopen ElementaryAddCommGroup\n\n/-- In fact $k$ is at most\n $$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\nlemma construct_good_improved' :\n k \u2264 \u03b4 + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 # T\u2081 | T\u2082] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2082 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2082 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2083 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2083 | T\u2082] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2081 | T\u2082] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2081 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2083 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2083 | T\u2082] - d[p.X\u2080\u2082 # X\u2082])) := by\n have I1 : I[T\u2082 : T\u2081] = I[T\u2081 : T\u2082] := mutualInfo_comm hT\u2082 hT\u2081 _\n have I2 : I[T\u2083 : T\u2081] = I[T\u2081 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2081 _\n have I3 : I[T\u2083 : T\u2082] = I[T\u2082 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2082 _\n have Z123 := construct_good_prelim' h_min hT hT\u2081 hT\u2082 hT\u2083\n have h132 : T\u2081 + T\u2083 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z132 := construct_good_prelim' h_min h132 hT\u2081 hT\u2083 hT\u2082\n have h213 : T\u2082 + T\u2081 + T\u2083 = 0 := by rw [\u2190 hT]; abel\n have Z213 := construct_good_prelim' h_min h213 hT\u2082 hT\u2081 hT\u2083\n have h231 : T\u2082 + T\u2083 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z231 := construct_good_prelim' h_min h231 hT\u2082 hT\u2083 hT\u2081\n have h312 : T\u2083 + T\u2081 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z312 := construct_good_prelim' h_min h312 hT\u2083 hT\u2081 hT\u2082\n have h321 : T\u2083 + T\u2082 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z321 := construct_good_prelim' h_min h321 hT\u2083 hT\u2082 hT\u2081\n simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321\n linarith\n\n/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\nlemma construct_good_improved'' {\u03a9' : Type*} [MeasurableSpace \u03a9'] (\u03bc : Measure \u03a9')\n [IsProbabilityMeasure \u03bc] {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G}\n (hT : T\u2081 + T\u2082 + T\u2083 = 0) (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083) :\n k \u2264 I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc] + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]))\n := by\n let M : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n exact construct_good_improved' h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend aux\n\nopen scoped BigOperators\n\n/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\nlemma averaged_construct_good : k \u2264 (I[U : V | S] + I[V : W | S] + I[W : U | S])\n + (p.\u03b7 / 6) * (((d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]))\n + ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])))\n := by\n have hS : Measurable S := ((hX\u2081.add' hX\u2082).add' hX\u2081').add' hX\u2082'\n have hU : Measurable U := hX\u2081.add' hX\u2082\n have hV : Measurable V := hX\u2081'.add hX\u2082\n have hW : Measurable W := hX\u2081'.add hX\u2081\n have hUVW : U + V + W = 0 := sum_uvw_eq_zero X\u2081 X\u2082 X\u2081'\n have hz (a : \u211d) : a = \u2211 z, (\u2119 (S \u207b\u00b9' {z})).toReal * a := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u2119 hS, one_mul]\n rw [hz k, hz (d[p.X\u2080\u2081 # X\u2081]), hz (d[p.X\u2080\u2082 # X\u2082])]\n simp only [condMutualInfo_eq_sum' hS, \u2190 Finset.sum_add_distrib, \u2190 mul_add,\n condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, \u2190 Finset.sum_sub_distrib, \u2190 mul_sub, Finset.mul_sum,\n \u2190 mul_assoc (p.\u03b7/6), mul_comm (p.\u03b7/6), mul_assoc _ _ (p.\u03b7/6)]\n rw [Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_le_sum (fun i _hi \u21a6 ?_)\n rcases eq_or_ne (\u2119 (S \u207b\u00b9' {i})) 0 with h'i|h'i\n \u00b7 simp [h'i]\n rw [mul_assoc, \u2190 mul_add]\n gcongr\n have : IsProbabilityMeasure (\u2119[|S \u207b\u00b9' {i}]) := cond_isProbabilityMeasure \u2119 h'i\n linarith [construct_good_improved'' h_min (\u2119[|S \u207b\u00b9' {i}]) hUVW hU hV hW]\n\nvariable (p)\n\nlemma dist_diff_bound_1 :\n (d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n \u2264 (16 * k + 6 * d[X\u2081 # X\u2081] + 2 * d[X\u2082 # X\u2082]) / 4 + (H[X\u2081 + X\u2081'] - H[X\u2082 + X\u2082']) / 4\n + (H[X\u2082 | X\u2082 + X\u2082'] - H[X\u2081 | X\u2081 + X\u2081']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2082' X\u2081' hX\u2081 hX\u2082 hX\u2082' hX\u2081' h_indep.reindex_four_abcd\n have I2 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep.reindex_four_abdc\n have I3 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2082 X\u2081' hX\u2081 hX\u2082' hX\u2082 hX\u2081' h_indep.reindex_four_acbd\n have I4 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2081' X\u2082 hX\u2081 hX\u2082' hX\u2081' hX\u2082 h_indep.reindex_four_acdb\n have I5 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082 X\u2082' hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_adbc\n have I6 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082' X\u2082 hX\u2081 hX\u2081' hX\u2082' hX\u2082 h_indep.reindex_four_adcb\n have C1 : U + X\u2082' + X\u2081' = S := by abel\n have C2 : W + X\u2082 + X\u2082' = S := by abel\n have C3 : X\u2081 + X\u2082' + X\u2082 + X\u2081' = S := by abel\n have C4 : X\u2081 + X\u2082' + X\u2081' + X\u2082 = S := by abel\n have C5 : W + X\u2082' + X\u2082 = S := by abel\n have C7 : X\u2082 + X\u2081' = V := by abel\n have C8 : X\u2081 + X\u2081' = W := by abel\n have C9 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082.symm\n have C10 : d[X\u2082 # X\u2081'] = d[X\u2081' # X\u2082] := rdist_symm\n have C11 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081.symm\n have C12 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq h\u2082.symm\n have C13 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] := (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082.symm\n have C14 : d[X\u2081' # X\u2082] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C15 : H[X\u2081' + X\u2082'] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C16 : H[X\u2082'] = H[X\u2082] := h\u2082.symm.entropy_eq\n have C17 : H[X\u2081'] = H[X\u2081] := h\u2081.symm.entropy_eq\n have C18 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C19 : H[X\u2082' + X\u2081'] = H[U] := by rw [add_comm]; exact C15\n have C20 : d[X\u2082' # X\u2082] = d[X\u2082 # X\u2082] := h\u2082.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C21 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C22 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C23 : X\u2082' + X\u2082 = X\u2082 + X\u2082' := by abel\n have C24 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C25 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C27 : H[X\u2082' | X\u2082 + X\u2082'] = H[X\u2082 | X\u2082 + X\u2082'] := by\n conv_lhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2082'.add' hX\u2082) hX\u2082 (hX\u2082.add' hX\u2082')\n have I : IdentDistrib (\u27e8X\u2082', X\u2082\u27e9) (\u27e8X\u2082, X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2082.symm h\u2082\n (h_indep.indepFun (show 2 \u2260 1 by decide)) (h_indep.indepFun (show 1 \u2260 2 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C28 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C29 : H[X\u2081' | V] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082) hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C30 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n simp only [C1, C2, C3, C4, C5, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30] at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma dist_diff_bound_2 :\n ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]))\n \u2264 (16 * k + 6 * d[X\u2082 # X\u2082] + 2 * d[X\u2081 # X\u2081]) / 4 + (H[X\u2082 + X\u2082'] - H[X\u2081 + X\u2081']) / 4\n + (H[X\u2081 | X\u2081 + X\u2081'] - H[X\u2082 | X\u2082 + X\u2082']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2082' X\u2081' hX\u2082 hX\u2081 hX\u2082' hX\u2081' h_indep.reindex_four_bacd\n have I2 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2081' X\u2082' hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_badc\n have I3 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081 X\u2081' hX\u2082 hX\u2082' hX\u2081 hX\u2081' h_indep.reindex_four_bcad\n have I4 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081' X\u2081 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bcda\n have I5 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2081 X\u2082' hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bdac\n have I6 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2082' X\u2081 hX\u2082 hX\u2081' hX\u2082' hX\u2081 h_indep.reindex_four_bdca\n have C1 : X\u2082 + X\u2081 = X\u2081 + X\u2082 := by abel\n have C2 : X\u2081 + X\u2081' = W := by abel\n have C3 : U + X\u2082' + X\u2081' = S := by abel\n have C4 : X\u2082 + X\u2081' = V := by abel\n have C5 : X\u2082 + X\u2082' + X\u2081 + X\u2081' = S := by abel\n have C6 : X\u2082 + X\u2082' + X\u2081' + X\u2081 = S := by abel\n have C7 : V + X\u2081 + X\u2082' = S := by abel\n have C8 : V + X\u2082' + X\u2081 = S := by abel\n have C9 : d[X\u2082 # X\u2081] = d[X\u2081 # X\u2082] := rdist_symm\n have C10 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n have C11 : d[X\u2082 # X\u2081'] = d[X\u2081 # X\u2082] := by\n rw [rdist_symm]\n exact ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n have C12 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C13 : d[X\u2082' # X\u2081] = d[X\u2081 # X\u2082'] := rdist_symm\n have C14 : d[X\u2081' # X\u2081] = d[X\u2081 # X\u2081'] := rdist_symm\n have C15 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n have C16 : H[X\u2081' + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C17 : H[X\u2082' + X\u2081'] = H[X\u2081 + X\u2082] := by rw [add_comm]; exact C16\n have C18 : H[X\u2081'] = H[X\u2081] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2081.symm\n have C19 : H[X\u2082'] = H[X\u2082] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2082.symm\n have C20 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C21 : H[X\u2081' | W] = H[X\u2081 | W] := by\n conv_rhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2081) hX\u2081 (hX\u2081.add' hX\u2081')\n have I : IdentDistrib (\u27e8X\u2081', X\u2081\u27e9) (\u27e8X\u2081, X\u2081'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2081\n (h_indep.indepFun (show 3 \u2260 0 by decide)) (h_indep.indepFun (show 0 \u2260 3 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C22 : H[X\u2082' | X\u2082' + X\u2081] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C23 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C24 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C25 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C27 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n have C28 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C29 : H[X\u2082' + X\u2081] = H[X\u2081 + X\u2082] := by\n rw [add_comm]\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C30 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2081.symm\n have C31 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable) h\u2082.symm\n simp only [C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C25, C26, C27, C28, C29, C30, C31]\n at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma averaged_final : k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) := by\n apply (averaged_construct_good hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_min).trans\n have : 0 \u2264 p.\u03b7 := p.h\u03b7.le\n have := sum_condMutual_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep.reindex_four_abdc h_min\n gcongr ?_ + (p.\u03b7 / 6) * ?_\n linarith [dist_diff_bound_1 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep,\n dist_diff_bound_2 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep]\n\n/-- Suppose $0 < \\eta < 1/8$. Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$. The proof\nof this lemma uses copies `X\u2081', X\u2082'` already in the context. For a version that does not assume\nthese are given and constructs them instead, use `tau_strictly_decreases'`.\n-/\ntheorem tau_strictly_decreases_aux' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 := by\n have : 0 < p.\u03b7 := p.h\u03b7\n have : k \u2264 8 * p.\u03b7 * k := by calc\n k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) :=\n averaged_final p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n _ \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) +\n p.\u03b7 / 6 * (8 * k + 2 * (2 * (k + (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7)))) := by\n gcongr\n exact second_estimate_aux p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082\n h_indep.reindex_four_abdc h_min\n _ = 8 * p.\u03b7 * k - (1 - 5 * p.\u03b7 - 4/6 * p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7) := by ring\n _ \u2264 8 * p.\u03b7 * k := by\n simp only [tsub_le_iff_right, le_add_iff_nonneg_right]\n apply div_nonneg _ (by linarith)\n apply mul_nonneg (by linarith) _\n linarith [first_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min]\n apply le_antisymm _ (rdist_nonneg hX\u2081 hX\u2082)\n nlinarith\n\ntheorem tau_strictly_decreases' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 := by\n let \u27e8A, mA, \u03bc, Y\u2081, Y\u2082, Y\u2081', Y\u2082', h\u03bc, h_indep, hY\u2081, hY\u2082, hY\u2081', hY\u2082', h_id1, h_id2, h_id1', h_id2'\u27e9\n := independent_copies4_nondep hX\u2081 hX\u2082 hX\u2081 hX\u2082 \u2119 \u2119 \u2119 \u2119\n rw [\u2190 h_id1.rdist_eq h_id2]\n letI : MeasureSpace A := \u27e8\u03bc\u27e9\n have : IsProbabilityMeasure (\u2119 : Measure A) := h\u03bc\n rw [\u2190 h_id1.tau_minimizes p h_id2] at h_min\n exact tau_strictly_decreases_aux' p hY\u2081 hY\u2082 hY\u2081' hY\u2082' (h_id1.trans h_id1'.symm)\n (h_id2.trans h_id2'.symm) h_indep.reindex_four_abdc h_min hp\n\nend MainEstimates\n\nsection EntropicPFR\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nopen scoped Topology\nopen Filter Set\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n [MeasurableSingletonClass G]\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\n/-- For `p.\u03b7 \u2264 1/8`, there exist \u03c4-minimizers `X\u2081, X\u2082` at zero Rusza distance. For `p.\u03b7 < 1/8`,\nall minimizers are fine, by `tau_strictly_decreases'`. For `p.\u03b7 = 1/8`, we use a limit of\nminimizers for `\u03b7 < 1/8`, which exists by compactness. -/\nlemma tau_minimizer_exists_rdist_eq_zero :\n \u2203 (\u03a9 : Type uG) (m\u03a9 : MeasureSpace \u03a9) (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G),\n Measurable X\u2081 \u2227 Measurable X\u2082 \u2227 IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 tau_minimizes p X\u2081 X\u2082\n \u2227 d[X\u2081 # X\u2082] = 0 := by\n -- let `u\u2099` be a sequence converging from below to `\u03b7`. In particular, `u\u2099 < 1/8`.\n obtain \u27e8u, -, u_mem, u_lim\u27e9 :\n \u2203 u, StrictMono u \u2227 (\u2200 (n : \u2115), u n \u2208 Set.Ioo 0 p.\u03b7) \u2227 Tendsto u atTop (\ud835\udcdd p.\u03b7) :=\n exists_seq_strictMono_tendsto' p.h\u03b7\n -- For each `n`, consider a minimizer associated to `\u03b7 = u\u2099`.\n let q : \u2115 \u2192 refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G := fun n \u21a6\n \u27e8p.X\u2080\u2081, p.X\u2080\u2082, p.hmeas1, p.hmeas2, u n, (u_mem n).1, by linarith [(u_mem n).2, p.h\u03b7']\u27e9\n have : \u2200 n, \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | q n] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | q n] :=\n fun n \u21a6 tau_min_exists_measure (q n)\n choose \u03bc \u03bc1_prob \u03bc2_prob h\u03bc using this\n -- The minimizer associated to `u\u2099` is at zero Rusza distance of itself, by\n -- lemma `tau_strictly_decreases'`.\n have I n : d[id ; (\u03bc n).1 # id ; (\u03bc n).2] = 0 := by\n let M : MeasureSpace (G \u00d7 G) := \u27e8(\u03bc n).1.prod (\u03bc n).2\u27e9\n have : IsProbabilityMeasure ((\u03bc n).1.prod (\u03bc n).2) := by infer_instance\n have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; (\u03bc n).1 # id ; (\u03bc n).2] :=\n IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id\n rw [\u2190 this]\n apply tau_strictly_decreases' (q n) measurable_fst measurable_snd ?_\n (by linarith [(u_mem n).2, p.h\u03b7'])\n intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | q n] = \u03c4[id ; (\u03bc n).1 # id ; (\u03bc n).2 | q n] :=\n ProbabilityTheory.IdentDistrib.tau_eq (q n) IdentDistrib.fst_id IdentDistrib.snd_id\n rw [A]\n exact h\u03bc n _ _ h\u2081 h\u2082\n -- extract a converging subsequence of the sequence of minimizers, seen as pairs of probability\n -- measures on `G` (which is a compact space).\n let \u03bc' : \u2115 \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun n \u21a6 (\u27e8(\u03bc n).1, \u03bc1_prob n\u27e9, \u27e8(\u03bc n).2, \u03bc2_prob n\u27e9)\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G)\n have : DiscreteTopology G := \u27e8rfl\u27e9\n -- The limiting pair of measures will be the desired minimizer.\n rcases IsCompact.tendsto_subseq (x := \u03bc') isCompact_univ (fun n \u21a6 mem_univ _)\n with \u27e8\u03bd, -, \u03c6, \u03c6mono, h\u03bd\u27e9\n have \u03c6lim : Tendsto \u03c6 atTop atTop := \u03c6mono.tendsto_atTop\n let M : MeasureSpace (G \u00d7 G) := \u27e8(\u03bd.1 : Measure G).prod \u03bd.2\u27e9\n have P : IsProbabilityMeasure ((\u03bd.1 : Measure G).prod (\u03bd.2 : Measure G)) := by infer_instance\n refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_, ?_\u27e9\n -- check that it is indeed a minimizer, as a limit of minimizers.\n \u00b7 intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bd.1 # id ; \u03bd.2 | p] :=\n ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n rw [A]\n have L1 : Tendsto (fun n \u21a6 \u03c4[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2 | q (\u03c6 n)]) atTop\n (\ud835\udcdd (\u03c4[id ; \u03bd.1 # id ; \u03bd.2 | p])) := by\n apply Tendsto.add (Tendsto.add ?_ (Tendsto.mul (u_lim.comp \u03c6lim) ?_))\n (Tendsto.mul (u_lim.comp \u03c6lim) ?_)\n \u00b7 apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) h\u03bd\n \u00b7 have : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n apply Tendsto.comp (this.tendsto _) h\u03bd\n \u00b7 have : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n apply Tendsto.comp (this.tendsto _) h\u03bd\n have L2 : Tendsto (fun n \u21a6 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | q (\u03c6 n)]) atTop\n (\ud835\udcdd (\u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p])) :=\n Tendsto.add (Tendsto.add tendsto_const_nhds (Tendsto.mul (u_lim.comp \u03c6lim)\n tendsto_const_nhds)) (Tendsto.mul (u_lim.comp \u03c6lim) tendsto_const_nhds)\n exact le_of_tendsto_of_tendsto' L1 L2 (fun n \u21a6 h\u03bc (\u03c6 n) _ _ h\u2081 h\u2082)\n -- check that it has zero Rusza distance, as a limit of a sequence at zero Rusza distance.\n \u00b7 have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; \u03bd.1 # id ; \u03bd.2] :=\n IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id\n rw [this]\n have L1 : Tendsto (fun n \u21a6 d[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2]) atTop\n (\ud835\udcdd (d[id ; \u03bd.1 # id ; (\u03bd.2 : Measure G)])) := by\n apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) h\u03bd\n have L2 : Tendsto (fun n \u21a6 d[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2]) atTop (\ud835\udcdd 0) := by simp [I]\n exact tendsto_nhds_unique L1 L2\n\n/-- `entropic_PFR_conjecture_improv`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\n subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture_improv (hp\u03b7 : p.\u03b7 = 1/8) :\n \u2203 H : AddSubgroup G, \u2203 \u03a9 : Type uG, \u2203 m\u03a9 : MeasureSpace \u03a9, \u2203 U : \u03a9 \u2192 G,\n IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 Measurable U \u2227\n IsUniform H U \u2227 d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 10 * d[p.X\u2080\u2081 # p.X\u2080\u2082] := by\n obtain \u27e8\u03a9', m\u03a9', X\u2081, X\u2082, hX\u2081, hX\u2082, hP, htau_min, hdist\u27e9 := tau_minimizer_exists_rdist_eq_zero p\n obtain \u27e8H, U, hU, hH_unif, hdistX\u2081, hdistX\u2082\u27e9 := exists_isUniform_of_rdist_eq_zero hX\u2081 hX\u2082 hdist\n refine \u27e8H, \u03a9', inferInstance, U, inferInstance, hU, hH_unif , ?_\u27e9\n have h : \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[p.X\u2080\u2082 # p.X\u2080\u2081 | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1\n rw [tau, tau, hp\u03b7] at h\n norm_num at h\n have : d[p.X\u2080\u2081 # p.X\u2080\u2082] = d[p.X\u2080\u2082 # p.X\u2080\u2081] := rdist_symm\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # X\u2081] + d[X\u2081 # U] := rdist_triangle p.hmeas1 hX\u2081 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # X\u2082] + d[X\u2082 # U] := rdist_triangle p.hmeas2 hX\u2082 hU\n linarith\n\n/-- `entropic_PFR_conjecture_improv'`: For two $G$-valued random variables $X^0_1, X^0_2$, there is\nsome subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$., and\nd[X^0_1; U_H] and d[X^0_2; U_H] are at most 5/2 * d[X^0_1;X^0_2] -/\ntheorem entropic_PFR_conjecture_improv' (hp\u03b7 : p.\u03b7 = 1/8) :\n \u2203 H : AddSubgroup G, \u2203 \u03a9 : Type uG, \u2203 m\u03a9 : MeasureSpace \u03a9, \u2203 U : \u03a9 \u2192 G,\n IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 Measurable U \u2227\n IsUniform H U \u2227 d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 10 * d[p.X\u2080\u2081 # p.X\u2080\u2082] \u2227 d[p.X\u2080\u2081 # U] \u2264 11/2 * d[p.X\u2080\u2081 # p.X\u2080\u2082] \u2227 d[p.X\u2080\u2082 # U] \u2264 11/2 * d[p.X\u2080\u2081 # p.X\u2080\u2082] := by\n obtain \u27e8\u03a9', m\u03a9', X\u2081, X\u2082, hX\u2081, hX\u2082, hP, htau_min, hdist\u27e9 := tau_minimizer_exists_rdist_eq_zero p\n obtain \u27e8H, U, hU, hH_unif, hdistX\u2081, hdistX\u2082\u27e9 := exists_isUniform_of_rdist_eq_zero hX\u2081 hX\u2082 hdist\n have : d[p.X\u2080\u2081 # p.X\u2080\u2082] = d[p.X\u2080\u2082 # p.X\u2080\u2081] := rdist_symm\n have goal\u2081 : d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 10 * d[p.X\u2080\u2081 # p.X\u2080\u2082] := by\n have h : \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[p.X\u2080\u2082 # p.X\u2080\u2081 | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1\n rw [tau, tau, hp\u03b7] at h\n norm_num at h\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # X\u2081] + d[X\u2081 # U] := rdist_triangle p.hmeas1 hX\u2081 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # X\u2082] + d[X\u2082 # U] := rdist_triangle p.hmeas2 hX\u2082 hU\n linarith\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # p.X\u2080\u2082] + d[p.X\u2080\u2082 # U] := rdist_triangle p.hmeas1 p.hmeas2 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # p.X\u2080\u2081] + d[p.X\u2080\u2081 # U] := rdist_triangle p.hmeas2 p.hmeas1 hU\n refine \u27e8H, \u03a9', inferInstance, U, inferInstance, hU, hH_unif, goal\u2081, by linarith, by linarith\u27e9\n\n\n\n\n\n\n\n\n\nend EntropicPFR\n\nsection PFR\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\nopen Pointwise Set MeasureTheory ProbabilityTheory Real Fintype Function\n\nvariable {G \u03a9 : Type*} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G]\n {A B : Set G} {K : \u211d}\n\n", "theoremStatement": "/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of\nan elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$\nsuch that $A$ can be covered by at most $K^6 |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^10$. -/\nlemma PFR_conjecture_improv_aux (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c \u2264 K ^ 6 * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n \u2227 Nat.card H \u2264 K ^ 10 * Nat.card A \u2227 Nat.card A \u2264 K ^ 10 * Nat.card H \u2227 A \u2286 c + H ", "theoremName": "PFR_conjecture_improv_aux", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "d7b6033604422555584766a86b127ec46025a905", "date": "2023-12-15"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 859, "tokenPositionInFile": 46915, "theoremPositionInFile": 17}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 33, "numPremises": 477, "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", 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"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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n classical\n let mG : MeasurableSpace G := \u22a4\n have : MeasurableSingletonClass G := \u27e8\u03bb _ \u21a6 trivial\u27e9\n obtain \u27e8A_pos, -, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K :=\n PFR_conjecture_pos_aux' h\u2080A hA\n let A' := A.toFinite.toFinset\n have h\u2080A' : Finset.Nonempty A' := by\n simp [A', Finset.Nonempty]\n exact h\u2080A\n have hAA' : A' = A := Finite.coe_toFinset (toFinite A)\n rcases exists_isUniform_measureSpace A' h\u2080A' with \u27e8\u03a9\u2080, m\u03a9\u2080, UA, hP\u2080, UAmeas, UAunif, -\u27e9\n rw [hAA'] at UAunif\n have hadd_sub : A + A = A - A := by\n rw [\u2190 Set.image2_add, \u2190 Set.image2_sub]\n congr! 1 with a _ b _\n rw [(show a+b=a-b by simp)]\n rfl\n rw [hadd_sub] at hA\n have : d[UA # UA] \u2264 log K := rdist_le_of_isUniform_of_card_add_le h\u2080A hA UAunif UAmeas\n rw [\u2190 hadd_sub] at hA\n\n let p : refPackage \u03a9\u2080 \u03a9\u2080 G := \u27e8UA, UA, UAmeas, UAmeas, 1/8, (by norm_num), (by norm_num)\u27e9\n -- entropic PFR gives a subgroup `H` which is close to `A` for the Rusza distance\n rcases entropic_PFR_conjecture_improv p (by norm_num)\n with \u27e8H, \u03a9\u2081, m\u03a9\u2081, UH, hP\u2081, UHmeas, UHunif, hUH\u27e9\n rcases independent_copies_two UAmeas UHmeas\n with \u27e8\u03a9, m\u03a9, VA, VH, hP, VAmeas, VHmeas, Vindep, idVA, idVH\u27e9\n have VAunif : IsUniform A VA := UAunif.of_identDistrib idVA.symm $ measurableSet_discrete _\n have VA'unif := VAunif\n rw [\u2190 hAA'] at VA'unif\n have VHunif : IsUniform H VH := UHunif.of_identDistrib idVH.symm $ measurableSet_discrete _\n let H' := (H:Set G).toFinite.toFinset\n have hHH' : H' = (H:Set G) := Finite.coe_toFinset (toFinite (H:Set G))\n have VH'unif := VHunif\n rw [\u2190 hHH'] at VH'unif\n\n have : d[VA # VH] \u2264 10/2 * log K := by rw [idVA.rdist_eq idVH]; linarith\n have H_pos : (0 : \u211d) < Nat.card (H : Set G) := by\n have : 0 < Nat.card (H : Set G) := Nat.card_pos\n positivity\n have VA_ent : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' VAunif VAmeas\n have VH_ent : H[VH] = log (Nat.card (H : Set G)) := IsUniform.entropy_eq' VHunif VHmeas\n have Icard : |log (Nat.card A) - log (Nat.card (H : Set G))| \u2264 10 * log K := by\n rw [\u2190 VA_ent, \u2190 VH_ent]\n apply (diff_ent_le_rdist VAmeas VHmeas).trans\n linarith\n have IAH : Nat.card A \u2264 K ^ 10 * Nat.card (H : Set G) := by\n have : log (Nat.card A) \u2264 log K * 10 + log (Nat.card (H : Set G)) := by\n linarith [(le_abs_self _).trans Icard]\n convert exp_monotone this using 1\n \u00b7 exact (exp_log A_pos).symm\n \u00b7 rw [exp_add, exp_log H_pos, \u2190 rpow_def_of_pos K_pos]\n have IHA : Nat.card (H : Set G) \u2264 K ^ 10 * Nat.card A := by\n have : log (Nat.card (H : Set G)) \u2264 log K * 10 + log (Nat.card A) := by\n linarith [(neg_le_abs _).trans Icard]\n convert exp_monotone this using 1\n \u00b7 exact (exp_log H_pos).symm\n \u00b7 rw [exp_add, exp_log A_pos, \u2190 rpow_def_of_pos K_pos]\n -- entropic PFR shows that the entropy of `VA - VH` is small\n have I : log K * (-10/2) + log (Nat.card A) * (-1/2) + log (Nat.card (H : Set G)) * (-1/2)\n \u2264 - H[VA - VH] := by\n rw [Vindep.rdist_eq VAmeas VHmeas] at this\n have : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' VAunif VAmeas\n have : H[VH] = log (Nat.card (H : Set G)) := IsUniform.entropy_eq' VHunif VHmeas\n linarith\n -- therefore, there exists a point `x\u2080` which is attained by `VA - VH` with a large probability\n obtain \u27e8x\u2080, h\u2080\u27e9 : \u2203 x\u2080 : G, rexp (- H[VA - VH]) \u2264 (\u2119 : Measure \u03a9).real ((VA - VH) \u207b\u00b9' {x\u2080}) :=\n prob_ge_exp_neg_entropy' _ ((VAmeas.sub VHmeas).comp measurable_id')\n -- massage the previous inequality to get that `A \u2229 (H + {x\u2080})` is large\n have J : K ^ (-10/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (1/2) \u2264\n Nat.card (A \u2229 (H + {x\u2080}) : Set G) := by\n rw [VA'unif.measureReal_preimage_sub VAmeas VH'unif VHmeas Vindep] at h\u2080\n have := (Real.exp_monotone I).trans h\u2080\n have hAA'_card : Nat.card A' = Nat.card A := congrArg Nat.card (congrArg Subtype hAA')\n have hHH'_card : Nat.card H' = Nat.card (H : Set G) := congrArg Nat.card (congrArg Subtype hHH')\n rw [hAA'_card, hHH'_card, le_div_iff] at this\n convert this using 1\n . rw [exp_add, exp_add, \u2190 rpow_def_of_pos K_pos, \u2190 rpow_def_of_pos A_pos, \u2190 rpow_def_of_pos H_pos]\n rpow_ring\n norm_num\n . rw [hAA', hHH']\n positivity\n\n have Hne : Set.Nonempty (A \u2229 (H + {x\u2080} : Set G)) := by\n by_contra h'\n have : (0 : \u211d) < Nat.card (A \u2229 (H + {x\u2080}) : Set G) := lt_of_lt_of_le (by positivity) J\n simp only [Nat.card_eq_fintype_card, card_of_isEmpty, CharP.cast_eq_zero, lt_self_iff_false,\n not_nonempty_iff_eq_empty.1 h'] at this\n /- use Rusza covering lemma to cover `A` by few translates of `A \u2229 (H + {x\u2080}) - A \u2229 (H + {x\u2080})`\n (which is contained in `H`). The number of translates is at most\n `#(A + (A \u2229 (H + {x\u2080}))) / #(A \u2229 (H + {x\u2080}))`, where the numerator is controlled as this is\n a subset of `A + A`, and the denominator is bounded below by the previous inequality`. -/\n rcases Set.exists_subset_add_sub (toFinite A) (toFinite (A \u2229 ((H + {x\u2080} : Set G)))) Hne with\n \u27e8u, hu, Au, -\u27e9\n have Iu : Nat.card u \u2264 K ^ 6 * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := by\n have : (0 : \u211d) \u2264 Nat.card u := by simp\n have Z1 := mul_le_mul_of_nonneg_left J this\n have Z2 : (Nat.card u * Nat.card (A \u2229 (H + {x\u2080}) : Set G) : \u211d)\n \u2264 Nat.card (A + A \u2229 (\u2191H + {x\u2080})) := by norm_cast\n have Z3 : (Nat.card (A + A \u2229 (\u2191H + {x\u2080})) : \u211d) \u2264 K * Nat.card A := by\n apply le_trans _ hA\n simp only [Nat.cast_le]\n apply Nat.card_mono (toFinite _)\n apply add_subset_add_left (inter_subset_left _ _)\n have : 0 \u2264 K ^ (10/2) * Nat.card A ^ (-1/2) * Nat.card (H : Set G) ^ (-1/2) := by positivity\n have T := mul_le_mul_of_nonneg_left ((Z1.trans Z2).trans Z3) this\n convert T using 1 <;> rpow_ring <;> norm_num\n have A_subset_uH : A \u2286 u + H := by\n apply Au.trans\n rw [add_sub_assoc]\n apply add_subset_add_left\n apply (sub_subset_sub (inter_subset_right _ _) (inter_subset_right _ _)).trans\n rintro - \u27e8-, \u27e8y, hy, xy, hxy, rfl\u27e9, -, \u27e8z, hz, xz, hxz, rfl\u27e9, rfl\u27e9\n simp only [mem_singleton_iff] at hxy hxz\n simpa [hxy, hxz, -ElementaryAddCommGroup.sub_eq_add] using H.sub_mem hy hz\n exact \u27e8H, u, Iu, IHA, IAH, A_subset_uH\u27e9", "proofType": "tactic", "proofLengthLines": 116, "proofLengthTokens": 6141}} +{"srcContext": "import PFR.FirstEstimate\nimport PFR.SecondEstimate\nimport PFR.ForMathlib.Elementary\n\n/-!\n# Endgame\n\nThe endgame on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ be independent random variables, with $X_1,\\tilde X_1$ copies\n of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $d[X_1;X_2] = k$\n* $U := X_1 + X_2$\n* $V := \\tilde X_1 + X_2$\n* $W := X_1 + \\tilde X_1$\n* $S := X_1 + X_2 + \\tilde X_1 + \\tilde X_2$.\n* $I_1 := I[U : V | S]$\n* $I_2 := I[U : W | S]$\n* $I_3 := I[V : W | S]$ (not explicitly defined in Lean)\n\n# Main results:\n\n* `sum_condMutual_le` : An upper bound on the total conditional mutual information $I_1+I_2+I_3$.\n* `sum_dist_diff_le`: A sum of the \"costs\" of $U$, $V$, $W$.\n* `construct_good`: A construction of two random variables with small Ruzsa distance between them\n given some random variables with control on total cost, as well as total mutual information.\n-/\n\nopen MeasureTheory ProbabilityTheory\nopen scoped BigOperators\n\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [m\u03a9 : MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2081', X\u2082'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[U : V | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `V = X\u2081' + X\u2082`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[U : V | S]\n\n/-- `I\u2082 := I[U : W | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `W = X\u2081' + X\u2081`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[U : W | S]\n\n--(Mantas) this times out in the proof below\nprivate lemma hmeas2 :\n Measurable fun p : Fin 4 \u2192 G => ((p 0 + p 1, p 0 + p 2), p 0 + p 1 + p 2 + p 3) := by\n apply Measurable.prod\n \u00b7 apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n\n/-- The quantity $I_3 = I[V:W|S]$ is equal to $I_2$. -/\nlemma I\u2083_eq : I[V : W | S] = I\u2082 := by\n have h_indep2 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081, X\u2082'] := by\n exact h_indep.reindex_four_cbad\n have hident : IdentDistrib (fun a (i : Fin 4) => ![X\u2081, X\u2082, X\u2081', X\u2082'] i a)\n (fun a (j : Fin 4) => ![X\u2081', X\u2082, X\u2081, X\u2082'] j a) := by\n exact { aemeasurable_fst := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n aemeasurable_snd := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n map_eq := by\n rw [\u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081, X\u2082, X\u2081', X\u2082'])\n (Fin.cases hX\u2081 <| Fin.cases hX\u2082 <| Fin.cases hX\u2081' <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep,\n \u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081', X\u2082, X\u2081, X\u2082'])\n (Fin.cases hX\u2081' <| Fin.cases hX\u2082 <| Fin.cases hX\u2081 <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep2]\n congr\n ext i\n fin_cases i\n all_goals simp [h\u2081.map_eq] }\n have hmeas1 : Measurable (fun p : Fin 4 \u2192 G => (p 0 + p 1, p 0 + p 1 + p 2 + p 3)) := by\n simp_all only [Matrix.cons_val', Matrix.empty_val', Matrix.cons_val_fin_one]\n apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n have hUVS : IdentDistrib (prod U S) (prod V S) := by\n convert (IdentDistrib.comp hident hmeas1); simp; abel\n have hUVWS : IdentDistrib (prod (prod U W) S) (prod (prod V W) S) := by\n convert (IdentDistrib.comp hident hmeas2) <;> simp <;> abel\n have hU : Measurable U := Measurable.add hX\u2081 hX\u2082\n have hV : Measurable V := Measurable.add hX\u2081' hX\u2082\n have hW : Measurable W := Measurable.add hX\u2081' hX\u2081\n have hS : Measurable S := by measurability\n rw [condMutualInfo_eq hV hW hS, condMutualInfo_eq hU hW hS, chain_rule'' \u2119 hU hS,\n chain_rule'' \u2119 hV hS, chain_rule'' \u2119 hW hS, chain_rule'' \u2119 _ hS, chain_rule'' \u2119 _ hS,\n IdentDistrib.entropy_eq hUVS, IdentDistrib.entropy_eq hUVWS]\n \u00b7 exact Measurable.prod (by exact hU) (by exact hW)\n \u00b7 exact Measurable.prod (by exact hV) (by exact hW)\n\n\n/--\n$$ I(U : V | S) + I(V : W | S) + I(W : U | S) $$\nis less than or equal to\n$$ 6 \\eta k - \\frac{1 - 5 \\eta}{1-\\eta} (2 \\eta k - I_1).$$\n-/\nlemma sum_condMutual_le :\n I[U : V | S] + I[V : W | S] + I[W : U | S]\n \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) := by\n have : I[W:U|S] = I\u2082 := by\n rw [condMutualInfo_comm]\n \u00b7 exact Measurable.add' hX\u2081' hX\u2081\n \u00b7 exact Measurable.add' hX\u2081 hX\u2082\n rw [I\u2083_eq, this]\n have h\u2082 := second_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n have h := add_le_add (add_le_add_left h\u2082 I\u2081) h\u2082\n convert h using 1\n have : 1 - p.\u03b7 > 0 := by linarith [p.h\u03b7']\n field_simp [this]\n ring\n all_goals { simpa }\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" => d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nlemma hU : H[U] = H[X\u2081' + X\u2082'] :=\n IdentDistrib.entropy_eq (ProbabilityTheory.IdentDistrib.add h\u2081 h\u2082\n (iIndepFun.indepFun h_indep (show (0 : Fin 4) \u2260 1 by norm_cast))\n (iIndepFun.indepFun h_indep (show (2 : Fin 4) \u2260 3 by norm_cast)))\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition1 : iIndepFun (fun _ \u21a6 hG) ![X\u2081, X\u2082, X\u2081' + X\u2082'] :=\n h_indep.apply_two_last hX\u2081 hX\u2082 hX\u2081' hX\u2082' measurable_add\n\nlemma hV : H[V] = H[X\u2081 + X\u2082'] :=\nIdentDistrib.entropy_eq (ProbabilityTheory.IdentDistrib.add h\u2081.symm h\u2082\n (iIndepFun.indepFun h_indep (show (2 : Fin 4) \u2260 1 by norm_cast))\n (iIndepFun.indepFun h_indep (show (0 : Fin 4) \u2260 3 by norm_cast)))\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition2 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2081, X\u2081' + X\u2082'] :=\n independenceCondition1 hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_bacd\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition3 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081 + X\u2082'] :=\n independenceCondition1 hX\u2081' hX\u2082 hX\u2081 hX\u2082' h_indep.reindex_four_cbad\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition4 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2081', X\u2081 + X\u2082'] :=\n independenceCondition1 hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bcad\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition5 : iIndepFun (fun _ \u21a6 hG) ![X\u2081, X\u2081', X\u2082 + X\u2082'] :=\n independenceCondition1 hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_acbd\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition6 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2082', X\u2081' + X\u2081] :=\n independenceCondition1 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bdca\n\nset_option maxHeartbeats 400000 in\n/--\n$$ \\sum_{i=1}^2 \\sum_{A\\in\\{U,V,W\\}} \\big(d[X^0_i;A|S] - d[X^0_i;X_i]\\big)$$\nis less than or equal to\n$$ \\leq (6 - 3\\eta) k + 3(2 \\eta k - I_1).$$\n-/\nlemma sum_dist_diff_le :\n c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 (6 - 3 * p.\u03b7)*k + 3 * (2*p.\u03b7*k - I\u2081) := by\n let X\u2080\u2081 := p.X\u2080\u2081\n let X\u2080\u2082 := p.X\u2080\u2082\n have ineq1 : d[X\u2080\u2081 # U | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 := by\n have aux1 : H[S] + H[U] - H[X\u2081] - H[X\u2081' + X\u2082'] = H[S] - H[X\u2081] := by\n rw [hU X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep]; ring\n have aux2 : d[X\u2080\u2081 # U | U + (X\u2081' + X\u2082')] - d[X\u2080\u2081 # X\u2081]\n \u2264 (H[U + (X\u2081' + X\u2082')] + H[U] - H[X\u2081] - H[X\u2081' + X\u2082']) / 2 :=\n condRuzsaDist_diff_ofsum_le \u2119 (hX := p.hmeas1) (hY := hX\u2081) (hZ := hX\u2082)\n (Measurable.add hX\u2081' hX\u2082') (independenceCondition1 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n rw [\u2190 add_assoc, aux1] at aux2\n linarith [aux2]\n\n have ineq2 : d[X\u2080\u2082 # U | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := by\n have aux1 : H[S] + H[U] - H[X\u2082] - H[X\u2081' + X\u2082'] = H[S] - H[X\u2082] := by\n rw [hU X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep] ; ring\n have aux2 : d[X\u2080\u2082 # U | U + (X\u2081' + X\u2082')] - d[X\u2080\u2082 # X\u2082]\n \u2264 (H[U + (X\u2081' + X\u2082')] + H[U] - H[X\u2082] - H[X\u2081' + X\u2082']) / 2 := by\n rw [(show U = X\u2082 + X\u2081 from add_comm _ _)]\n apply condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas2) (hX\u2082) (hX\u2081)\n (Measurable.add hX\u2081' hX\u2082') (independenceCondition2 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n rw [\u2190 add_assoc, aux1] at aux2\n linarith [aux2]\n\n have V_add_eq : V + (X\u2081 + X\u2082') = S := by abel\n\n have ineq3 : d[X\u2080\u2081 # V | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 := by\n have aux2 : d[p.X\u2080\u2081 # V | V + (X\u2081 + X\u2082')] - d[p.X\u2080\u2081 # X\u2081']\n \u2264 (H[V + (X\u2081 + X\u2082')] + H[V] - H[X\u2081'] - H[X\u2081 + X\u2082']) / 2 :=\n condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas1) (hX\u2081') (hX\u2082) (Measurable.add hX\u2081 hX\u2082')\n (independenceCondition3 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n\n have aux1 : H[S] + H[V] - H[X\u2081'] - H[X\u2081 + X\u2082'] = H[S ; \u2119] - H[X\u2081 ; \u2119] := by\n rw [hV X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep, h\u2081.entropy_eq]; ring\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl p.hmeas1.aemeasurable) h\u2081,\n V_add_eq, aux1] at aux2\n linarith [aux2]\n\n have ineq4 : d[X\u2080\u2082 # V | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := by\n have aux2 : d[p.X\u2080\u2082 # V | V + (X\u2081 + X\u2082')] - d[p.X\u2080\u2082 # X\u2082]\n \u2264 (H[V + (X\u2081 + X\u2082')] + H[V] - H[X\u2082] - H[X\u2081 + X\u2082']) / 2 := by\n rw [(show V = X\u2082 + X\u2081' from add_comm _ _)]\n apply condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas2) (hX\u2082) (hX\u2081') (Measurable.add hX\u2081 hX\u2082')\n (independenceCondition4 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have aux1 : H[S] + H[V] - H[X\u2082] - H[X\u2081 + X\u2082'] = H[S ; \u2119] - H[X\u2082 ; \u2119] := by\n rw [hV X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep]; ring\n rw [V_add_eq, aux1] at aux2\n linarith [aux2]\n\n let W' := X\u2082 + X\u2082'\n have ineq5 : d[X\u2080\u2081 # W | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] + H[W ; \u2119] - H[X\u2081 ; \u2119] - H[W' ; \u2119])/2 := by\n have := condRuzsaDist_diff_ofsum_le \u2119 p.hmeas1 hX\u2081 hX\u2081' (Measurable.add hX\u2082 hX\u2082')\n (independenceCondition5 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have S_eq : X\u2081 + X\u2081' + (fun a \u21a6 X\u2082 a + X\u2082' a) = S := by\n rw [(show (fun a \u21a6 X\u2082 a + X\u2082' a) = X\u2082 + X\u2082' by rfl), \u2190 add_assoc, add_assoc X\u2081, add_comm X\u2081',\n \u2190 add_assoc]\n rwa [S_eq, add_comm X\u2081 X\u2081'] at this\n\n have ineq6 : d[X\u2080\u2082 # W' | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] + H[W' ; \u2119] - H[X\u2082 ; \u2119] - H[W ; \u2119])/2 := by\n have := condRuzsaDist_diff_ofsum_le \u2119 p.hmeas2 hX\u2082 hX\u2082' (Measurable.add hX\u2081' hX\u2081)\n (independenceCondition6 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have S_eq : X\u2082 + X\u2082' + (fun a \u21a6 X\u2081' a + X\u2081 a) = S := by\n rw [(show (fun a \u21a6 X\u2081' a + X\u2081 a) = X\u2081' + X\u2081 by rfl), add_comm, \u2190 add_assoc, add_comm X\u2081',\n add_assoc X\u2081, add_comm X\u2081', \u2190 add_assoc]\n rwa [S_eq] at this\n\n have dist_eq : d[X\u2080\u2082 # W' | S] = d[X\u2080\u2082 # W | S] := by\n have S_eq : S = (X\u2082 + X\u2082') + (X\u2081' + X\u2081) := by\n rw [add_comm X\u2081' X\u2081, add_assoc _ X\u2082', add_comm X\u2082', \u2190 add_assoc X\u2082, \u2190 add_assoc X\u2082,\n add_comm X\u2082]\n rw [S_eq]\n apply condRuzsaDist'_of_inj_map' p.hmeas2 (hX\u2082.add hX\u2082') (hX\u2081'.add hX\u2081)\n\n -- Put everything together to bound the sum of the `c` terms\n have ineq7 : c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264\n 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := by\n have step\u2081 : c[U|S # U|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc\n _ = (d[p.X\u2080\u2081 # U|S] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2082 # U|S] - d[p.X\u2080\u2082 # X\u2082]) := by ring\n _ \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 + (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := add_le_add ineq1 ineq2\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n have step\u2082 : c[V|S # V|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc c[V|S # V|S] =(d[p.X\u2080\u2081 # V|S] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2082 # V|S] - d[p.X\u2080\u2082 # X\u2082]) := by ring\n _ \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 + (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := add_le_add ineq3 ineq4\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n have step\u2083 : c[W|S # W|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc c[W|S # W|S] = (d[X\u2080\u2081 # W | S] - d[X\u2080\u2081 # X\u2081]) + (d[X\u2080\u2082 # W' | S] - d[X\u2080\u2082 # X\u2082]) :=\n by rw [dist_eq]\n _ \u2264 (H[S ; \u2119] + H[W ; \u2119] - H[X\u2081 ; \u2119] - H[W' ; \u2119])/2 + (H[S ; \u2119] + H[W' ; \u2119] - H[X\u2082 ; \u2119] - H[W ; \u2119])/2\n := add_le_add ineq5 ineq6\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) +\n (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) + (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) :=\n add_le_add (add_le_add step\u2081 step\u2082) step\u2083\n _ = 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := by ring\n\n have h_indep' : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'] := by\n apply ProbabilityTheory.iIndepFun.reindex (Equiv.swap (2 : Fin 4) 3)\n convert h_indep using 1\n ext x\n fin_cases x ; all_goals { aesop }\n\n have ineq8 : 3 * H[S ; \u2119] \u2264 3/2 * (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081 :=\n calc 3 * H[S ; \u2119] \u2264 3 * (H[X\u2081 ; \u2119] / 2 + H[X\u2082 ; \u2119] / 2 + (2+p.\u03b7)*k - I\u2081) := by\n apply (mul_le_mul_left (zero_lt_three' \u211d)).mpr\n (ent_ofsum_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep' h_min)\n _ = 3/2 * ( H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081 := by ring\n\n -- Final computation\n calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := ineq7\n _ = 3 * H[S ; \u2119] - (3/2 *(H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])) := by ring\n _ \u2264 (3/2 * ( H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081) - (3/2 *(H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])) :=\n sub_le_sub_right ineq8 _\n _ = (6 - 3 * p.\u03b7)*k + 3 * (2*p.\u03b7*k - I\u2081) := by ring\n\n/-- $U+V+W=0$. -/\nlemma sum_uvw_eq_zero : U+V+W = 0 := by\n rw [add_comm X\u2081' X\u2082, ElementaryAddCommGroup.sum_add_sum_add_sum_eq_zero]\n\nsection construct_good\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081+T\u2082+T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n", "theoremStatement": "/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\nThen there exist random variables $T'_1, T'_2$ such that\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X_1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2]) $$\nis at most\n$$ \\delta + \\eta ( d[X^0_1;T_1]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2]-d[X^0_2;X_2]) $$\n$$ + \\tfrac12 \\eta I[T_1: T_3] + \\tfrac12 \\eta I[T_2: T_3].$$\n-/\nlemma construct_good_prelim :\n k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 # T\u2082] + p.\u03b7 * (I[T\u2081: T\u2083] + I[T\u2082 : T\u2083])/2 ", "theoremName": "construct_good_prelim", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "6660391f1b64de611e0ddad38e36e42940b01b0f", "date": "2023-11-24"}, "file": "PFR/PFR/Endgame.lean", "module": "PFR.Endgame", "jsonFile": "PFR.Endgame.jsonl", "positionMetadata": {"lineInFile": 346, "tokenPositionInFile": 15327, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": false, 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"Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n\n have hp.\u03b7 : 0 \u2264 p.\u03b7 := by linarith [p.h\u03b7]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 :=\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have h2T\u2081 : T\u2081 = T\u2082 + T\u2083 := by simp [h2T\u2083, add_left_comm]\n have h2T\u2082 : T\u2082 = T\u2083 + T\u2081 := by simp [h2T\u2081, add_left_comm]\n\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n\n have h2 : p.\u03b7 * sum2 \u2264 p.\u03b7 * (d[p.X\u2080\u2081 # T\u2081] - d[p.X\u2080\u2081 # X\u2081] + I[T\u2081 : T\u2083] / 2) := by\n have : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), sum2]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n gcongr\n linarith [condRuzsaDist_le' \u2119 \u2119 p.hmeas1 hT\u2081 hT\u2083]\n\n have h3 : p.\u03b7 * sum3 \u2264 p.\u03b7 * (d[p.X\u2080\u2082 # T\u2082] - d[p.X\u2080\u2082 # X\u2082] + I[T\u2082 : T\u2083] / 2) := by\n have : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp [sum3, integral_sub (.of_finite _ _) (.of_finite _ _)]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n gcongr\n linarith [condRuzsaDist_le' \u2119 \u2119 p.hmeas2 hT\u2082 hT\u2083]\n\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * c[T\u2081 # T\u2082] + p.\u03b7 * (I[T\u2081 : T\u2083] + I[T\u2082 : T\u2083]) / 2 := by\n suffices sum4 = sum1 + p.\u03b7 * (sum2 + sum3) by linarith\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _), integral_mul_left]\n\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $ ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4", "proofType": "tactic", "proofLengthLines": 51, "proofLengthTokens": 2763}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\n", "theoremStatement": "lemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 ", "theoremName": "gen_ineq_aux2", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "7b9dc64d2818953df55b83e56a95ceac2d68457e", "date": "2023-12-13"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 74, "tokenPositionInFile": 3088, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 26, "numPremises": 466, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "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.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", 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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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring", "proofType": "tactic", "proofLengthLines": 116, "proofLengthTokens": 6694}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] := by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]\n\n/-- Adding a constant to a random variable does not change the Rusza distance. -/\nlemma rdist_add_const [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y + fun _ \u21a6 c; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n have A : IdentDistrib (Y' + fun _ \u21a6 c) (Y + fun _ \u21a6 c) \u03bd \u03bc' := by\n change IdentDistrib (fun \u03c9 \u21a6 Y' \u03c9 + c) (fun \u03c9 \u21a6 Y \u03c9 + c) \u03bd \u03bc'\n apply hIdY.comp (measurable_add_const c)\n have B : IndepFun X' (Y' + fun _ \u21a6 c) \u03bd := by\n change IndepFun X' (fun \u03c9 \u21a6 Y' \u03c9 + c) \u03bd\n apply hind.comp measurable_id (measurable_add_const c)\n have C : X' - (Y' + fun _ \u21a6 c) = (X' - Y') + (fun _ \u21a6 -c) := by\n ext \u03c9; simp; abel\n rw [\u2190 hIdX.rdist_eq hIdY, \u2190 hIdX.rdist_eq A, hind.rdist_eq hX' hY',\n B.rdist_eq hX' (hY'.add_const _), entropy_add_const hY' c, C, entropy_add_const]\n exact hX'.sub hY'\n\n/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\nlemma rdist_add_const' [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] (c: G) (c': G)\n (hX : Measurable X) (hY : Measurable Y) :\n d[X + fun _ \u21a6 c; \u03bc # Y + fun _ \u21a6 c'; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_add_const _ hY, rdist_symm, rdist_add_const hY hX, rdist_symm]\n measurability\n\n/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (X : \u03a9 \u2192 G) (Y : \u03a9 \u2192 G) (Z : \u03a9 \u2192 G)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n have h1 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] + H[X - Y; \u03bc] \u2264 H[\u27e8X - Z, X - Y\u27e9; \u03bc] + H[\u27e8Y, X - Y\u27e9; \u03bc] :=\n entropy_triple_add_entropy_le \u03bc (hX.sub hZ) hY (hX.sub hY)\n have h2 : H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n calc H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n have : \u27e8X - Z, X - Y\u27e9 = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 \u27e8X - Z, Y - Z\u27e9 := by ext1; simp\n rw [this]\n apply entropy_comp_le \u03bc (by measurability)\n _ \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n have h : 0 \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] - H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n apply mutualInfo_nonneg (by measurability) (by measurability) \u03bc\n linarith\n have h3 : H[\u27e8Y, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X, Y\u27e9 ; \u03bc] := by\n have : \u27e8Y, X - Y\u27e9 = (fun p \u21a6 (p.2, p.1 - p.2)) \u2218 \u27e8X, Y\u27e9 := by ext1; simp\n rw [this]\n exact entropy_comp_le \u03bc (hX.prod_mk hY) _\n have h4 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] = H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] := by\n refine entropy_of_comp_eq_of_comp \u03bc ((hX.sub hZ).prod_mk (hY.prod_mk (hX.sub hY)))\n (hX.prod_mk (hY.prod_mk hZ))\n (fun p : G \u00d7 (G \u00d7 G) \u21a6 (p.2.2 + p.2.1, p.2.1, -p.1 + p.2.2 + p.2.1))\n (fun p : G \u00d7 G \u00d7 G \u21a6 (p.1 - p.2.2, p.2.1, p.1 - p.2.1)) ?_ ?_\n \u00b7 ext1; simp\n \u00b7 ext1; simp\n have h5 : H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [entropy_assoc hX hY hZ, entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h\n rw [h4, h5] at h1\n calc H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Y - Z; \u03bc] - H[Z; \u03bc] := by linarith\n _ = H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n congr 2\n rw [entropy_sub_comm hY hZ]\n\n/-- The **entropic Ruzsa triangle inequality** -/\nlemma rdist_triangle {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [h\u03bc : IsProbabilityMeasure \u03bc] [h\u03bc' : IsProbabilityMeasure \u03bc'] [h\u03bc'' : IsProbabilityMeasure \u03bc''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n d[X ; \u03bc # Z ; \u03bc''] \u2264 d[X ; \u03bc # Y ; \u03bc'] + d[Y ; \u03bc' # Z ; \u03bc''] := by\n obtain \u27e8A, mA, \u03bcA, X', Y', Z', h\u03bcA, hInd, hX', hY', hZ', HX, HY, HZ, _, _, _\u27e9 :=\n independent_copies3_nondep_finiteRange hX hY hZ \u03bc \u03bc' \u03bc''\n suffices d[X' ; \u03bcA # Z' ; \u03bcA] \u2264 d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] by\n rwa [HX.rdist_eq HY, HY.rdist_eq HZ, HX.rdist_eq HZ] at this\n have IndepLem : IndepFun (\u27e8X', Z'\u27e9) Y' \u03bcA :=\n iIndepFun.indepFun_prod_mk hInd (fun i => by fin_cases i ; all_goals { simpa }) 0 2 1\n (by norm_cast) (by norm_cast)\n calc d[X' ; \u03bcA # Z' ; \u03bcA] = H[X' - Z'; \u03bcA] - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) := by\n rw [ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using hInd.indepFun (show 0 \u2260 2 by norm_cast)) hX' hZ'] ; ring\n _ \u2264 (H[X' - Y' ; \u03bcA] + H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA]) - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) :=\n sub_le_sub_right (ent_of_diff_le _ _ _ hX' hZ' hY' IndepLem) _\n _ = (H[X' - Y' ; \u03bcA] - H[X'; \u03bcA] / 2 - H[Y' ; \u03bcA] / 2) +\n (H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA] / 2 - H[Z'; \u03bcA] / 2) := by ring\n _ = d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] := by\n rw [ProbabilityTheory.IndepFun.rdist_eq (by simpa using (iIndepFun.indepFun hInd\n (show 0 \u2260 1 by norm_cast))) hX' hY', ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using (iIndepFun.indepFun hInd (show 1 \u2260 2 by norm_cast))) hY' hZ']\n\nvariable [MeasurableSingletonClass S] [MeasurableSingletonClass T]\n\n/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) [IsFiniteMeasure \u03bc]\n (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc'\"]\" => condRuzsaDist X Z Y W \u03bc \u03bc'\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" # \" Y \" | \" W \"]\" => condRuzsaDist X Z Y W volume volume\n\nlemma condRuzsaDist_def (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W] := rfl\n\n/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\nlemma condRuzsaDist_symm {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hZ : Measurable Z)\n (hW : Measurable W) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange Z]\n [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[Y | W ; \u03bc' # X | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist_symm]\n\n@[simp] lemma condRuszaDist_zero_right (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n d[X | Z ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero,\n kernel.rdist_zero_right]\n\n@[simp] lemma condRuszaDist_zero_left (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; 0 # Y | W ; \u03bc'] = 0 := by\n simp [condRuzsaDist]\n\nlemma condRuzsaDist_nonneg {X : \u03a9 \u2192 G} (hX : Measurable X) [FiniteRange X]\n {Z : \u03a9 \u2192 S} (hZ : Measurable Z) [FiniteRange Z]\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) [FiniteRange Y]\n {W : \u03a9' \u2192 T} (hW : Measurable W) [FiniteRange W]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n 0 \u2264 d[X | Z ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def]\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n refine kernel.rdist_nonneg ?_ ?_\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hX hZ\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hY hW\n\n/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\nlemma rdist_eq_rdistm : d[X ; \u03bc # Y ; \u03bc'] = kernel.rdistm (\u03bc.map X) (\u03bc'.map Y) := rfl\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\nlemma condRuzsaDist_eq_sum {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z in FiniteRange.toFinset Z, \u2211 w in FiniteRange.toFinset W,\n (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((((FiniteRange.toFinset Z)\n \u00d7\u02e2 (FiniteRange.toFinset W)) : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Finset.sum_product,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\nlemma condRuzsaDist_eq_sum' {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [Fintype S] [Fintype T] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z, \u2211 w, (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Fintype.sum_prod_type,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist' (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc' \"]\" => condRuzsaDist' X Y W \u03bc \u03bc'\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" # \" Y \" | \" W \"]\" => condRuzsaDist' X Y W volume volume\n\n/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\nlemma condRuzsaDist'_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y | W ; \u03bc'] =\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W] :=\n rfl\n\n@[simp] lemma condRuzsaDist'_zero_right (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) :\n d[X ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist'_def, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero, kernel.rdist_zero_right]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\nlemma condRuzsaDist'_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (dirac ()) (\u03bc'.map W) ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 FiniteRange.toFinset W : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, smul_eq_mul, Finset.sum_product]\n simp only [Finset.univ_unique, PUnit.default_eq_unit, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, Pi.one_apply, one_mul,\n Finset.sum_singleton]\n simp_rw [Measure.map_apply hW (measurableSet_singleton _)]\n congr with w\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff]\n refine Or.inr ?_\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm, condDistrib_apply hY hW _ _ hw]\n congr\n\n/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\nlemma condRuzsaDist'_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_eq_sum hY hW \u03bc \u03bc']\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\nopen scoped ENNReal\n\nlemma condRuzsaDist'_prod_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [FiniteRange W] [FiniteRange W']:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n have : d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc'] = \u2211 w in ((FiniteRange.toFinset W') \u00d7\u02e2 FiniteRange.toFinset W), (\u03bc' ((fun a => (W' a, W a)) \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; \u03bc'[|(fun a => (W' a, W a)) \u207b\u00b9' {w}]] := by\n rw [condRuzsaDist'_eq_sum hY (hW'.prod_mk hW)]\n apply Finset.sum_subset\n . intro (t, t')\n simp\n intro \u03c9 h1 h2\n exact \u27e8\u27e8\u03c9, h1\u27e9, \u27e8\u03c9, h2\u27e9\u27e9\n intro (t, t') _ ht\n simp at ht\n have : (fun a => (W' a, W a)) \u207b\u00b9' {(t, t')} = \u2205 := by\n ext \u03c9\n simp\n exact ht \u03c9\n simp [this]\n rw [this, Finset.sum_product_right]\n congr 1 with w\n rw [condRuzsaDist'_eq_sum hY hW', Finset.mul_sum]\n congr 1 with w'\n rw [\u2190 mul_assoc]\n have A : (fun a \u21a6 (W' a, W a)) \u207b\u00b9' {(w', w)} = W' \u207b\u00b9' {w'} \u2229 W\u207b\u00b9' {w} := by ext; simp\n congr 1\n \u00b7 simp only [ProbabilityTheory.cond, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n smul_eq_mul, ENNReal.toReal_mul, A, restrict_apply (hW' (MeasurableSet.singleton w'))]\n rcases eq_bot_or_bot_lt (\u03bc' (W \u207b\u00b9' {w})) with hw|hw\n \u00b7 have : \u03bc' (W' \u207b\u00b9' {w'} \u2229 W \u207b\u00b9' {w}) = 0 :=\n le_antisymm (le_trans (measure_mono (Set.inter_subset_right _ _)) hw.le) bot_le\n simp [hw, this]\n \u00b7 rw [\u2190 mul_assoc, \u2190 ENNReal.toReal_mul, ENNReal.mul_inv_cancel, ENNReal.one_toReal, one_mul]\n exacts [hw.ne', by finiteness]\n \u00b7 congr 1\n rw [A, cond_cond_eq_cond_inter' _ (hW (.singleton w)) (hW' (.singleton w')), Set.inter_comm]\n finiteness\n\n/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\nlemma condRuzsaDist'_prod_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [Fintype T]:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_prod_eq_sum \u03bc \u03bc' hY hW' hW]\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\nlemma condRuzsaDist'_eq_integral (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = (\u03bc'.map W)[fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])]] := by\n rw [condRuzsaDist'_eq_sum hY hW]\n simp_rw [\u2190 smul_eq_mul]\n have : (\u03bc'.map W) (FiniteRange.toFinset W : Set T)\u1d9c = 0 := by\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n convert symm $ integral_eq_sum' (\u03bc'.map W) this (fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])])\n rw [Measure.map_apply hW (MeasurableSet.singleton _)]\n\n/-- Conditioning by a constant does not affect Ruzsa distance. -/\nlemma condRuzsaDist_of_const {X : \u03a9 \u2192 G} (hX : Measurable X) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (c : S)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange W] :\n d[X|(fun _ \u21a6 c) ; \u03bc # Y | W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def, condRuzsaDist'_def, Measure.map_const,measure_univ,one_smul, kernel.rdist,\n kernel.rdist, integral_prod, integral_dirac, integral_prod,integral_dirac]\n dsimp; congr; ext x; congr\n rw [condDistrib_apply hX measurable_const]\n \u00b7 simp\n \u00b7 simp\n \u00b7 exact integrable_of_finiteSupport _\n \u00b7 exact integrable_of_finiteSupport _\n\n/-- If $(X,Z)$ and $(Y,W)$ are independent, then\n$$ d[X | Z ; Y | W] = H[X'- Y'|Z', W'] - H[X'|Z']/2 - H[Y'|W']/2$$\n-/\nlemma condRuzsaDist_of_indep\n {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun (\u27e8X, Z\u27e9) (\u27e8Y, W\u27e9) \u03bc) [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc] = H[X - Y | \u27e8Z, W\u27e9 ; \u03bc] - H[X | Z ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ (hZ.prod_mk hW),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hW]\n swap; \u00b7 exact hX.sub hY\n congr 2\n have hZW : IndepFun Z W \u03bc := by\n have h' := IndepFun.comp h measurable_snd measurable_snd\n exact h'\n have hZW_map : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map Z).prod (\u03bc.map W) :=\n (indepFun_iff_map_prod_eq_prod_map_map hZ.aemeasurable hW.aemeasurable).mp hZW\n rw [\u2190 hZW_map]\n refine kernel.entropy_congr ?_\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (hZ.prod_mk hW) _ _).symm\n refine (this.symm.trans ?_).symm\n suffices kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft S (condDistrib Y W \u03bc)\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc by\n filter_upwards [this] with x hx\n rw [kernel.map_apply, kernel.map_apply, hx]\n . exact (condDistrib_eq_prod_of_indepFun hX hZ hY hW \u03bc h).symm\n\n/-- Formula for conditional Ruzsa distance for independent sets of variables. -/\nlemma condRuzsaDist'_of_indep {X : \u03a9 \u2192 G} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun X (\u27e8Y, W\u27e9) \u03bc) [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc] = H[X - Y | W ; \u03bc] - H[X ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist'_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ hW,\n condEntropy_eq_kernel_entropy hY hW, entropy_eq_kernel_entropy]\n rotate_left\n \u00b7 exact hX.sub hY\n congr 2\n let Z : \u03a9 \u2192 Unit := fun _ \u21a6 ()\n rw [\u2190 condDistrib_unit_right hX \u03bc]\n have h' : IndepFun (\u27e8X,Z\u27e9) (\u27e8Y, W\u27e9) \u03bc := by\n rw [indepFun_iff_measure_inter_preimage_eq_mul]\n intro s t hs ht\n have : \u27e8X, Z\u27e9 \u207b\u00b9' s = X \u207b\u00b9' ((fun c \u21a6 (c, ())) \u207b\u00b9' s) := by ext1 y; simp\n rw [this]\n rw [indepFun_iff_measure_inter_preimage_eq_mul] at h\n exact h _ _ (measurable_prod_mk_right hs) ht\n have h_indep := condDistrib_eq_prod_of_indepFun hX measurable_const hY hW _ h'\n have h_meas_eq : \u03bc.map (\u27e8Z, W\u27e9) = (Measure.dirac ()).prod (\u03bc.map W) := by\n ext s hs\n rw [Measure.map_apply (measurable_const.prod_mk hW) hs, Measure.prod_apply hs, lintegral_dirac,\n Measure.map_apply hW (measurable_prod_mk_left hs)]\n congr\n rw [\u2190 h_meas_eq]\n have : kernel.map (kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft Unit (condDistrib Y W \u03bc)) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc)\n (fun x \u21a6 x.1 - x.2) measurable_sub := by\n filter_upwards [h_indep] with y hy\n conv_rhs => rw [kernel.map_apply, hy]\n rw [kernel.entropy_congr this]\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (measurable_const.prod_mk hW) _ _).symm\n rw [kernel.entropy_congr this]\n have h_meas : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map W).map (Prod.mk ()) := by\n ext s hs\n rw [Measure.map_apply measurable_prod_mk_left hs, h_meas_eq, Measure.prod_apply hs,\n lintegral_dirac]\n have h_ker : condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.prodMkLeft Unit (condDistrib (X - Y) W \u03bc) := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply (measurable_const.prod_mk hW) (measurableSet_singleton _)] at hx\n ext s hs\n have h_preimage_eq : (fun a \u21a6 (PUnit.unit, W a)) \u207b\u00b9' {x} = W \u207b\u00b9' {x.2} := by\n conv_lhs => rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod]\n ext1 y\n simp\n rw [kernel.prodMkLeft_apply, condDistrib_apply' _ (measurable_const.prod_mk hW) _ _ hx hs,\n condDistrib_apply' _ hW _ _ _ hs]\n rotate_left\n \u00b7 exact hX.sub hY\n \u00b7 convert hx\n exact h_preimage_eq.symm\n \u00b7 exact hX.sub hY\n congr\n rw [kernel.entropy_congr h_ker, h_meas, kernel.entropy_prodMkLeft_unit]\n\n/-- The conditional Ruzsa distance is unchanged if the sets of random variables are replaced with\ncopies. -/\nlemma condRuzsaDist_of_copy {X : \u03a9 \u2192 G} (hX : Measurable X) {Z : \u03a9 \u2192 S} (hZ : Measurable Z)\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) {W : \u03a9' \u2192 T} (hW : Measurable W)\n {X' : \u03a9'' \u2192 G} (hX' : Measurable X') {Z' : \u03a9'' \u2192 S} (hZ' : Measurable Z')\n {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib (\u27e8X, Z\u27e9) (\u27e8X', Z'\u27e9) \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange Z] [FiniteRange W] [FiniteRange Z'] [FiniteRange W'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[X' | Z' ; \u03bc'' # Y' | W' ; \u03bc'''] := by\n classical\n set A := (FiniteRange.toFinset Z) \u222a (FiniteRange.toFinset Z')\n set B := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n have hfull' : Measure.prod (\u03bc''.map Z') (\u03bc'''.map W') ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hZZ' : \u03bc.map Z = \u03bc''.map Z' := (h1.comp measurable_snd).map_eq\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hZZ', \u2190 hWW',\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hz : \u03bc (Z \u207b\u00b9' {x.1}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 have hZZ'x : \u03bc (Z \u207b\u00b9' {x.1}) = \u03bc'' (Z' \u207b\u00b9' {x.1}) := by\n have : \u03bc.map Z {x.1} = \u03bc''.map Z' {x.1} := by rw [hZZ']\n rwa [Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hZ' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hX hZ _ _ hz hs, condDistrib_apply' hX' hZ' _ _ _ hs]\n swap; \u00b7 rwa [hZZ'x] at hz\n congr\n have : \u03bc.map (\u27e8X, Z\u27e9) (s \u00d7\u02e2 {x.1}) = \u03bc''.map (\u27e8X', Z'\u27e9) (s \u00d7\u02e2 {x.1}) := by rw [h1.map_eq]\n rwa [Measure.map_apply (hX.prod_mk hZ) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hX'.prod_mk hZ') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 X' a) \u207b\u00b9' s)] at this\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this\n\nlemma condRuzsaDist'_of_copy (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} (hY : Measurable Y)\n {W : \u03a9' \u2192 T} (hW : Measurable W)\n (X' : \u03a9'' \u2192 G) {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib X X' \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange W] [FiniteRange W'] :\n d[X ; \u03bc # Y | W ; \u03bc'] = d[X' ; \u03bc'' # Y' | W' ; \u03bc'''] := by\n classical\n set A := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (dirac ()) (\u03bc'.map W)\n ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 A : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n simp only [A]\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n have hfull' : Measure.prod (dirac ()) (\u03bc'''.map W')\n ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 A : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n simp only [A]\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, condRuzsaDist'_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hWW',\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 rw [kernel.const_apply, kernel.const_apply, h1.map_eq]\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this\n\nlemma condRuszaDist_prod_eq_of_indepFun {\u03bc : Measure \u03a9} {\u03bc' : Measure \u03a9'} {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G}\n {W W' : \u03a9' \u2192 T} (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (hW' : Measurable W') (h : IndepFun (\u27e8Y, W\u27e9) W' \u03bc')\n [IsProbabilityMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | \u27e8W, W'\u27e9 ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist'_prod_eq_sum' _ _ hY hW hW']\n have : d[X ; \u03bc # Y | W ; \u03bc'] = \u2211 z, (\u03bc' (W' \u207b\u00b9' {z})).toReal * d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u03bc' hW', one_mul]\n rw [this]\n congr with w\n rcases eq_or_ne (\u03bc' (W' \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n congr 1\n apply condRuzsaDist'_of_copy _ hY hW _ hY hW (IdentDistrib.refl hX.aemeasurable)\n exact (h.identDistrib_cond (MeasurableSet.singleton w) (hY.prod_mk hW) hW' hw).symm\n\nvariable (\u03bc \u03bc') in\nlemma condRuzsaDist_comp_right {T' : Type*} [Fintype T] [Fintype T'] [MeasurableSpace T']\n [MeasurableSingletonClass T'] [IsFiniteMeasure \u03bc']\n (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (e : T \u2192 T')\n (hY : Measurable Y) (hW : Measurable W) (he : Measurable e)\n (h'e : Injective e) :\n d[X ; \u03bc # Y | e \u2218 W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist'_eq_sum' hY (he.comp hW), condRuzsaDist'_eq_sum' hY hW]\n simp [Set.preimage_comp]\n have A i : e \u207b\u00b9' {e i} = {i} := by ext x; simp [h'e.eq_iff]\n symm\n refine Fintype.sum_of_injective e h'e _ _ (fun i hi \u21a6 ?_) (by simp [A])\n suffices e \u207b\u00b9' {i} = \u2205 by simp [this]\n simpa [Set.eq_empty_iff_forall_not_mem] using hi\n\nlemma condRuzsaDist_of_inj_map {G' : Type*} [Countable G'] [AddCommGroup G']\n [MeasurableSpace G'] [MeasurableSingletonClass G'] [IsProbabilityMeasure \u03bc]\n (Y : Fin 4 \u2192 \u03a9 \u2192 G) (h_indep : IndepFun (\u27e8Y 0, Y 2\u27e9) (\u27e8Y 1, Y 3\u27e9) \u03bc)\n (h_meas : \u2200 i, Measurable (Y i)) (\u03c0 : G \u00d7 G \u2192+ G')\n (h\u03c0 : \u2200 (h : G), Function.Injective (fun g \u21a6 \u03c0 (g, h)))\n [FiniteRange (Y 2)] [FiniteRange (Y 3)] :\n d[\u03c0 \u2218 \u27e8Y 0, Y 2\u27e9 | Y 2 ; \u03bc # \u03c0 \u2218 \u27e8Y 1, Y 3\u27e9 | Y 3 ; \u03bc] = d[Y 0 | Y 2 ; \u03bc # Y 1 | Y 3 ; \u03bc] := by\n let f (h : G) (g : G) : G' := \u03c0 (g, h)\n let f' : G \u00d7 G \u2192 G \u2192 G' := fun (h1, h2) \u21a6 fun g \u21a6 \u03c0 (g, h1 - h2)\n have hf' (t : G \u00d7 G) : Function.Injective (f' t) := fun _ _ h \u21a6 h\u03c0 _ h\n let f'' : G \u00d7 G \u2192 G' \u00d7 G := fun (g, h) \u21a6 (\u03c0 (g, h), h)\n have hf'' : Measurable f'' := measurable_of_countable _\n have hm1 : Measurable (Y 0 - Y 1) := (h_meas 0).sub (h_meas 1)\n have hm2 : Measurable (\u27e8Y 2, Y 3\u27e9) := (h_meas 2).prod_mk (h_meas 3)\n rw [condRuzsaDist_of_indep (h_meas 0) (h_meas 2) (h_meas 1) (h_meas 3) \u03bc h_indep,\n condRuzsaDist_of_indep ((measurable_of_countable _).comp ((h_meas 0).prod_mk (h_meas 2)))\n (h_meas 2) ((measurable_of_countable _).comp ((h_meas 1).prod_mk (h_meas 3))) (h_meas 3) \u03bc\n (h_indep.comp hf'' hf''),\n \u2190 condEntropy_of_injective \u03bc hm1 hm2 f' hf', \u2190 map_comp_sub \u03c0,\n \u2190 condEntropy_of_injective \u03bc (h_meas 0) (h_meas 2) f h\u03c0,\n \u2190 condEntropy_of_injective \u03bc (h_meas 1) (h_meas 3) f h\u03c0]\n rfl\n\nlemma condRuzsaDist'_of_inj_map [IsProbabilityMeasure \u03bc] [elem: ElementaryAddCommGroup G 2]\n {X B C : \u03a9 \u2192 G}\n (hX : Measurable X) (hB : Measurable B) (hC : Measurable C)\n (h_indep : IndepFun X (\u27e8B, C\u27e9) \u03bc) [FiniteRange X] [FiniteRange B] [FiniteRange C] :\n d[X ; \u03bc # B | B + C ; \u03bc] = d[X ; \u03bc # C | B + C ; \u03bc] := by\n let \u03c0 : G \u00d7 G \u2192+ G :=\n { toFun := fun x \u21a6 x.2 - x.1\n map_zero' := by simp\n map_add' := fun a b \u21a6 by simp only [Prod.snd_add, Prod.fst_add,\n ElementaryAddCommGroup.sub_eq_add]; abel }\n let Y : Fin 4 \u2192 \u03a9 \u2192 G := ![-X, C, fun _ \u21a6 0, B + C]\n have _ : FiniteRange (Y 0) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 1) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 2) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 3) := by simp [Y]; infer_instance\n\n have hY_meas : \u2200 i, Measurable (Y i) := by\n intro i\n fin_cases i\n exacts [hX.neg, hC, measurable_const, hB.add hC]\n calc d[X ; \u03bc # B | B + C ; \u03bc]\n = d[X | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # B | B + C ; \u03bc] := by\n rw [condRuzsaDist_of_const hX _ _]\n _ = d[\u03c0 \u2218 \u27e8-X, fun _ : \u03a9 \u21a6 (0 : G)\u27e9 | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # \u03c0 \u2218 \u27e8C, B + C\u27e9 | B + C ; \u03bc] := by\n congr\n \u00b7 ext1 \u03c9; simp [\u03c0]\n \u00b7 ext1 \u03c9\n simp only [AddMonoidHom.coe_mk, ZeroHom.coe_mk, Function.comp_apply, Pi.add_apply, \u03c0]\n abel\n _ = d[\u03c0 \u2218 \u27e8Y 0, Y 2\u27e9 | Y 2 ; \u03bc # \u03c0 \u2218 \u27e8Y 1, Y 3\u27e9 | Y 3 ; \u03bc] := by congr\n _ = d[-X | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # C | B + C ; \u03bc] := by\n rw [condRuzsaDist_of_inj_map _ _ hY_meas \u03c0 (fun _ \u21a6 sub_right_injective)]\n \u00b7 congr\n \u00b7 have h1 : (\u27e8Y 0, Y 2\u27e9) = (fun x \u21a6 (-x, 0)) \u2218 X := by ext1 \u03c9; simp [Y]\n have h2 : (\u27e8Y 1, Y 3\u27e9) = (fun p \u21a6 (p.2, p.1 + p.2)) \u2218 (\u27e8B, C\u27e9) := by\n ext1 \u03c9;\n simp only [ElementaryAddCommGroup.neg_eq_self, Matrix.cons_val_one, Matrix.head_cons,\n Function.comp_apply, Prod.mk.injEq, Matrix.cons_val', Pi.add_apply, Matrix.empty_val',\n Matrix.cons_val_fin_one, true_and, Y]\n congr\n rw [h1, h2]\n refine h_indep.comp ?_ ?_\n \u00b7 exact measurable_neg.prod_mk measurable_const\n \u00b7 exact measurable_snd.prod_mk (measurable_fst.add measurable_snd)\n _ = d[-X ; \u03bc # C | B + C ; \u03bc] := by rw [condRuzsaDist_of_const]; exact hX.neg\n _ = d[X ; \u03bc # C | B + C ; \u03bc] := by -- because ElementaryAddCommGroup G 2\n congr\n simp\n\nlemma condRuzsaDist'_of_inj_map' [elem: ElementaryAddCommGroup G 2] [IsProbabilityMeasure \u03bc]\n [IsProbabilityMeasure \u03bc''] {A : \u03a9'' \u2192 G} {B C : \u03a9 \u2192 G} (hA : Measurable A) (hB : Measurable B)\n (hC : Measurable C) [FiniteRange A] [FiniteRange B] [FiniteRange C] :\n d[A ; \u03bc'' # B | B + C ; \u03bc] = d[A ; \u03bc'' # C | B + C ; \u03bc] := by\n -- we want to apply `condRuzsaDist'_of_inj_map'`, but for that all variables need to be in the same\n -- probability space\n let \u03a9' := \u03a9'' \u00d7 \u03a9\n set X\u2082' : \u03a9' \u2192 G := A \u2218 Prod.fst with hX\u2082'_def\n have hX\u2082' : Measurable X\u2082' := hA.comp measurable_fst\n let B' : \u03a9' \u2192 G := B \u2218 Prod.snd\n have hB' : Measurable B' := hB.comp measurable_snd\n let C' : \u03a9' \u2192 G := C \u2218 Prod.snd\n have hC' : Measurable C' := hC.comp measurable_snd\n let \u03bc' : Measure \u03a9' := Measure.prod \u03bc'' \u03bc\n haveI : IsProbabilityMeasure \u03bc' := by infer_instance\n -- h1 and h2 should be applications of a new lemma?\n have h1 : d[A ; \u03bc'' # B | B + C ; \u03bc] = d[X\u2082' ; \u03bc' # B' | B' + C' ; \u03bc'] := by\n refine condRuzsaDist'_of_copy A hB (by measurability) X\u2082' hB' (by measurability) ?_ ?_\n \u00b7 constructor\n \u00b7 exact hA.aemeasurable\n \u00b7 exact hX\u2082'.aemeasurable\n \u00b7 rw [\u2190 Measure.map_map hA measurable_fst]\n simp [\u03bc']\n \u00b7 constructor\n \u00b7 exact (hB.prod_mk (hB.add hC)).aemeasurable\n \u00b7 exact (hB'.prod_mk (hB'.add hC')).aemeasurable\n \u00b7 have : \u27e8B', B' + C'\u27e9 = (\u27e8B, B + C\u27e9) \u2218 Prod.snd := by ext1 _; simp [C', B']\n rw [this, \u2190 Measure.map_map _ measurable_snd]\n \u00b7 simp only [Measure.map_snd_prod, measure_univ, one_smul, \u03bc']\n \u00b7 exact hB.prod_mk (hB.add hC)\n have h2 : d[A ; \u03bc'' # C | B + C ; \u03bc] = d[X\u2082' ; \u03bc' # C' | B' + C' ; \u03bc'] := by\n apply condRuzsaDist'_of_copy _ hC (by measurability) X\u2082' hC' (by measurability) ?_ ?_\n \u00b7 constructor\n \u00b7 exact hA.aemeasurable\n \u00b7 exact hX\u2082'.aemeasurable\n \u00b7 rw [\u2190 Measure.map_map hA measurable_fst]\n simp [\u03bc']\n \u00b7 constructor\n \u00b7 exact (hC.prod_mk (hB.add hC)).aemeasurable\n \u00b7 exact (hC'.prod_mk (hB'.add hC')).aemeasurable\n \u00b7 have : \u27e8C', B' + C'\u27e9 = (\u27e8C, B + C\u27e9) \u2218 Prod.snd := by ext1 _; simp [B', C']\n rw [this, \u2190 Measure.map_map _ measurable_snd]\n \u00b7 simp only [Measure.map_snd_prod, measure_univ, one_smul, \u03bc']\n \u00b7 exact hC.prod_mk (hB.add hC)\n rw [h1, h2, condRuzsaDist'_of_inj_map hX\u2082' hB' hC']\n rw [indepFun_iff_map_prod_eq_prod_map_map hX\u2082'.aemeasurable (hB'.prod_mk hC').aemeasurable]\n have h_prod : (fun \u03c9 \u21a6 (X\u2082' \u03c9, prod B' C' \u03c9)) = Prod.map A (\u27e8B, C\u27e9) := by ext1; simp [B', C', X\u2082']\n have h_comp_snd : (fun a \u21a6 (B' a, C' a)) = (\u27e8B, C\u27e9) \u2218 Prod.snd := by ext1; simp [B', C']\n rw [h_prod, h_comp_snd, hX\u2082'_def, \u2190 Measure.map_map _ measurable_snd,\n \u2190 Measure.map_map _ measurable_fst, Measure.map_prod_map]\n rotate_left\n \u00b7 exact hA\n \u00b7 exact hB.prod_mk hC\n \u00b7 exact hA\n \u00b7 exact hB.prod_mk hC\n simp [\u03bc']\n\n/-- The **Kaimanovich-Vershik inequality**. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$ -/\nlemma kaimanovich_vershik {X Y Z : \u03a9 \u2192 G} (h : iIndepFun (fun _ \u21a6 hG) ![X, Y, Z] \u03bc)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [IsProbabilityMeasure \u03bc]\n [FiniteRange X] [FiniteRange Z] [FiniteRange Y] :\n H[X + Y + Z ; \u03bc] - H[X + Y ; \u03bc] \u2264 H[Y + Z ; \u03bc] - H[Y ; \u03bc] := by\n suffices (H[X ; \u03bc] + H[Y ; \u03bc] + H[Z ; \u03bc]) + H[X + Y + Z ; \u03bc]\n \u2264 (H[X ; \u03bc] + H[Y + Z ; \u03bc]) + (H[Z ; \u03bc] + H[X + Y ; \u03bc]) by linarith\n have : \u2200 (i : Fin 3), Measurable (![X, Y, Z] i) := fun i \u21a6 by fin_cases i <;> assumption\n convert entropy_triple_add_entropy_le _ hX hZ (show Measurable (X + (Y + Z)) by measurability)\n using 2\n . calc\n H[X ; \u03bc] + H[Y ; \u03bc] + H[Z ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [IndepFun.entropy_pair_eq_add hX hY]\n convert h.indepFun (show 0 \u2260 1 by decide)\n _ = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n rw [IndepFun.entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h.indepFun_prod_mk this 0 1 2 (by decide) (by decide)\n _ = H[\u27e8X, \u27e8Z , X + (Y + Z)\u27e9\u27e9 ; \u03bc] := by\n apply entropy_of_comp_eq_of_comp \u03bc (by measurability) (by measurability)\n (fun ((x, y), z) \u21a6 (x, z, x + y + z)) (fun (a, b, c) \u21a6 ((a, c - a - b), b))\n all_goals { funext \u03c9; dsimp [prod]; ext <;> dsimp; abel }\n . rw [add_assoc]\n . symm\n refine (entropy_add_right hX (by measurability) _).trans $\n IndepFun.entropy_pair_eq_add hX (by measurability) ?_\n exact h.indepFun_add_right this 0 1 2 (by decide) (by decide)\n \u00b7 rw [eq_comm, \u2190 add_assoc]\n refine (entropy_add_right' hZ (by measurability) _).trans $\n IndepFun.entropy_pair_eq_add hZ (by measurability) ?_\n exact h.indepFun_add_right this 2 0 1 (by decide) (by decide)\n\n/-- A version of the **Kaimanovich-Vershik inequality** with some variables negated. -/\nlemma kaimanovich_vershik' {X Y Z : \u03a9 \u2192 G} (h : iIndepFun (fun _ \u21a6 hG) ![X, Y, Z] \u03bc)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [IsProbabilityMeasure \u03bc]\n [FiniteRange X] [FiniteRange Z] [FiniteRange Y] :\n H[X - (Y + Z) ; \u03bc] - H[X - Y ; \u03bc] \u2264 H[Y + Z ; \u03bc] - H[Y ; \u03bc] := by\n rw [\u2190 entropy_neg (hY.add' hZ), \u2190 entropy_neg hY]\n simp_rw [sub_eq_add_neg, neg_add, \u2190 add_assoc]\n refine kaimanovich_vershik ?_ hX ?_ hZ.neg\n convert (h.neg 1).neg 2\n ext i; fin_cases i\n \u00b7 simp (discharger := decide)\n \u00b7 simp (discharger := decide)\n \u00b7 rw [\u2190 show \u2200 h : 2 < 3, (2 : Fin 3) = \u27e82, h\u27e9 by intro; rfl]\n simp (discharger := decide)\n . exact hY.neg\n\nsection BalogSzemerediGowers\n\n", "theoremStatement": "/-- The **entropic Balog-Szemer\u00e9di-Gowers inequality**. Let $A, B$ be $G$-valued random variables on\n$\\Omega$, and set $Z := A+B$. Then\n$$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]. $$\nTODO: remove the hypothesis of `Fintype G` from here and from `condIndep_copies'` -/\nlemma ent_bsg [IsProbabilityMeasure \u03bc] {A B : \u03a9 \u2192 G} (hA : Measurable A) (hB : Measurable B)\n [Fintype G] :\n (\u03bc.map (A + B))[fun z \u21a6 d[A ; \u03bc[|(A + B) \u207b\u00b9' {z}] # B ; \u03bc[|(A + B) \u207b\u00b9' {z}]]]\n \u2264 3 * I[A : B; \u03bc] + 2 * H[A + B ; \u03bc] - H[A ; \u03bc] - H[B ; \u03bc] ", "theoremName": "ent_bsg", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "3cc02d585af63c3382fed3084b0bbf84a234eee4", "date": "2023-11-15"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 1113, "tokenPositionInFile": 56651, "theoremPositionInFile": 58}, 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"Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let Z := A + B\n have hZ : Measurable Z := hA.add hB\n obtain \u27e8\u03a9', _, AB\u2081, AB\u2082, Z', \u03bd, _, hAB\u2081, hAB\u2082, hZ', hABZ, hABZ\u2081, hABZ\u2082, hZ\u2081, hZ\u2082\u27e9 :=\n condIndep_copies' (\u27e8A, B\u27e9) Z (hA.prod_mk hB) hZ \u03bc (fun (a, b) c \u21a6 c = a + b)\n (measurable_discrete _) (eventually_of_forall fun _ \u21a6 rfl)\n let A\u2081 := fun \u03c9 \u21a6 (AB\u2081 \u03c9).1\n let B\u2081 := fun \u03c9 \u21a6 (AB\u2081 \u03c9).2\n let A\u2082 := fun \u03c9 \u21a6 (AB\u2082 \u03c9).1\n let B\u2082 := fun \u03c9 \u21a6 (AB\u2082 \u03c9).2\n replace hZ\u2081 : Z' = A\u2081 + B\u2081 := funext hZ\u2081\n replace hZ\u2082 : Z' = A\u2082 + B\u2082 := funext hZ\u2082\n have hA\u2081 : Measurable A\u2081 := hAB\u2081.fst\n have hB\u2081 : Measurable B\u2081 := hAB\u2081.snd\n have hA\u2082 : Measurable A\u2082 := hAB\u2082.fst\n have hB\u2082 : Measurable B\u2082 := hAB\u2082.snd\n have hZZ' : IdentDistrib Z' Z \u03bd \u03bc := hABZ\u2081.comp measurable_snd\n have :=\n calc\n H[\u27e8A\u2081, \u27e8B\u2081, A\u2081 - B\u2082\u27e9\u27e9 ; \u03bd]\n = H[\u27e8\u27e8A\u2081, B\u2081\u27e9, \u27e8\u27e8A\u2082, B\u2082\u27e9, Z'\u27e9\u27e9 ; \u03bd] := entropy_of_comp_eq_of_comp _\n (hA\u2081.prod_mk $ hB\u2081.prod_mk $ hA\u2081.sub hB\u2082) (hAB\u2081.prod_mk $ hAB\u2082.prod_mk hZ')\n (fun (a, b, c) \u21a6 ((a, b), (b + c, a - c), a + b))\n (fun ((a, b), (_c, d), _e) \u21a6 (a, b, a - d))\n (by funext; simpa [sub_add_eq_add_sub, Prod.ext_iff, \u2190 hZ\u2081, hZ\u2082, two_nsmul, \u2190 add_sub_assoc,\n add_comm, eq_sub_iff_add_eq] using congr_fun (hZ\u2082.symm.trans hZ\u2081) _) rfl\n _ = H[\u27e8\u27e8A\u2081, B\u2081\u27e9, Z'\u27e9 ; \u03bd] + H[\u27e8\u27e8A\u2082, B\u2082\u27e9, Z'\u27e9 ; \u03bd] - H[Z' ; \u03bd] :=\n ent_of_cond_indep hAB\u2081 hAB\u2082 hZ' hABZ\n _ = 2 * H[\u27e8\u27e8A, B\u27e9, Z\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n rw [two_mul]\n congr 1\n congr 1 <;> exact IdentDistrib.entropy_eq \u2039_\u203a\n exact hZZ'.entropy_eq\n _ = 2 * H[\u27e8A, B\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n congr 2\n exact entropy_prod_comp (hA.prod_mk hB) _ fun x \u21a6 x.1 + x.2\n have :=\n calc\n H[\u27e8A\u2081, A\u2081 - B\u2082\u27e9 ; \u03bd]\n = H[\u27e8A\u2081, B\u2082\u27e9 ; \u03bd] := entropy_sub_right hA\u2081 hB\u2082 _\n _ \u2264 H[A\u2081 ; \u03bd] + H[B\u2082 ; \u03bd] := entropy_pair_le_add hA\u2081 hB\u2082 _\n _ = H[A ; \u03bc] + H[B ; \u03bc] := by\n congr 1\n exact (hABZ\u2081.comp measurable_fst.fst).entropy_eq\n exact (hABZ\u2082.comp measurable_fst.snd).entropy_eq\n have :=\n calc\n H[\u27e8B\u2081, A\u2081 - B\u2082\u27e9 ; \u03bd]\n = H[\u27e8A\u2082, B\u2081\u27e9 ; \u03bd] := by\n rw [entropy_comm hB\u2081 (show Measurable (A\u2081 - B\u2082) from hA\u2081.sub hB\u2082),\n \u2190 entropy_sub_left' hA\u2082 hB\u2081, sub_eq_sub_iff_add_eq_add.2 $ hZ\u2081.symm.trans hZ\u2082]\n _ \u2264 H[A\u2082 ; \u03bd] + H[B\u2081 ; \u03bd] := entropy_pair_le_add hA\u2082 hB\u2081 _\n _ = H[A ; \u03bc] + H[B ; \u03bc] := by\n congr 1\n exact (hABZ\u2082.comp measurable_fst.fst).entropy_eq\n exact (hABZ\u2081.comp measurable_fst.snd).entropy_eq\n have :=\n calc\n _ \u2264 _ := entropy_triple_add_entropy_le \u03bd hA\u2081 hB\u2081 (show Measurable (A\u2081 - B\u2082) from hA\u2081.sub hB\u2082)\n _ \u2264 _ := add_le_add \u2039_\u203a \u2039_\u203a\n have :=\n calc\n H[A\u2081 - B\u2082 | Z' ; \u03bd]\n \u2264 H[A\u2081 - B\u2082 ; \u03bd] := condEntropy_le_entropy _ (hA\u2081.sub hB\u2082) hZ'\n _ \u2264 _ := le_sub_iff_add_le'.2 \u2039_\u203a\n _ = 2 * I[A : B ; \u03bc] + H[Z ; \u03bc] := by\n rw [\u2039H[\u27e8A\u2081, \u27e8B\u2081, A\u2081 - B\u2082\u27e9\u27e9 ; \u03bd] = _\u203a, mutualInfo_def]; ring\n have hA\u2081Z :=\n calc\n H[A\u2081 | Z' ; \u03bd]\n _ = H[\u27e8A\u2081, B\u2081\u27e9 ; \u03bd] - H[Z' ; \u03bd] := by\n rw [chain_rule'', hZ\u2081, entropy_add_right, entropy_comm] <;> assumption\n _ = H[\u27e8A, B\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n congr 1\n exact (hABZ\u2081.comp measurable_fst).entropy_eq\n exact hZZ'.entropy_eq\n _ = H[A ; \u03bc] + H[B ; \u03bc] - I[A : B ; \u03bc] - H[Z ; \u03bc] := by\n rw [\u2190 entropy_add_entropy_sub_mutualInfo]\n have hB\u2082Z :=\n calc\n H[B\u2082 | Z' ; \u03bd]\n _ = H[\u27e8A\u2082, B\u2082\u27e9 ; \u03bd] - H[Z' ; \u03bd] := by\n rw [chain_rule'', hZ\u2082, entropy_add_right', entropy_comm] <;> assumption\n _ = H[\u27e8A, B\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n congr 1\n exact (hABZ\u2082.comp measurable_fst).entropy_eq\n exact hZZ'.entropy_eq\n _ = H[A ; \u03bc] + H[B ; \u03bc] - I[A : B ; \u03bc] - H[Z ; \u03bc] := by\n rw [\u2190 entropy_add_entropy_sub_mutualInfo]\n save\n calc\n (\u03bc.map Z)[fun z \u21a6 d[A ; \u03bc[|Z \u2190 z] # B ; \u03bc[|Z \u2190 z]]]\n = (\u03bd.map Z')[fun z \u21a6 d[A\u2081 ; \u03bd[|Z' \u2190 z] # B\u2082 ; \u03bd[|Z' \u2190 z]]] := by\n rw [hZZ'.map_eq]\n refine' integral_congr_ae $ eventually_of_forall fun z \u21a6 _\n have hAA\u2081 : IdentDistrib A\u2081 A (\u03bd[|Z' \u2190 z]) (\u03bc[|Z \u2190 z]) :=\n (hABZ\u2081.comp $ measurable_fst.fst.prod_mk measurable_snd).cond\n (measurableSet_singleton z) hZ' hZ\n have hBB\u2082 : IdentDistrib B\u2082 B (\u03bd[|Z' \u2190 z]) (\u03bc[|Z \u2190 z]) :=\n (hABZ\u2082.comp $ measurable_fst.snd.prod_mk measurable_snd).cond\n (measurableSet_discrete _) hZ' hZ\n dsimp (config := {zeta := false}) [rdist]\n rw [\u2190 hAA\u2081.entropy_eq, \u2190 hBB\u2082.entropy_eq, hAA\u2081.map_eq, hBB\u2082.map_eq]\n _ = (\u03bd.map Z')[fun z \u21a6\n H[A\u2081 - B\u2082 ; \u03bd[|Z' \u2190 z]] - H[A\u2081 ; \u03bd[|Z' \u2190 z]]/2 - H[B\u2082 ; \u03bd[|Z' \u2190 z]]/2] :=\n integral_congr_ae $ hABZ.mono fun z hz \u21a6\n (hz.comp measurable_fst measurable_snd).rdist_eq hA\u2081 hB\u2082\n _ = H[A\u2081 - B\u2082 | Z' ; \u03bd] - H[A\u2081 | Z' ; \u03bd] / 2 - H[B\u2082 | Z' ; \u03bd] / 2 := by\n rw [integral_sub, integral_sub, integral_div, integral_div]\n rfl\n all_goals exact .of_finite _ _\n _ \u2264 2 * I[A : B ; \u03bc] + H[Z ; \u03bc] - H[A\u2081 | Z' ; \u03bd] / 2 - H[B\u2082 | Z' ; \u03bd] / 2 :=\n sub_le_sub_right (sub_le_sub_right \u2039_\u203a _) _\n _ = _ := by rw [hA\u2081Z, hB\u2082Z]; ring", "proofType": "tactic", "proofLengthLines": 113, "proofLengthTokens": 5046}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\nopen ElementaryAddCommGroup\n\n/-- In fact $k$ is at most\n $$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\nlemma construct_good_improved' :\n k \u2264 \u03b4 + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 # T\u2081 | T\u2082] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2082 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2082 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2083 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2083 | T\u2082] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2081 | T\u2082] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2081 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2083 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2083 | T\u2082] - d[p.X\u2080\u2082 # X\u2082])) := by\n have I1 : I[T\u2082 : T\u2081] = I[T\u2081 : T\u2082] := mutualInfo_comm hT\u2082 hT\u2081 _\n have I2 : I[T\u2083 : T\u2081] = I[T\u2081 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2081 _\n have I3 : I[T\u2083 : T\u2082] = I[T\u2082 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2082 _\n have Z123 := construct_good_prelim' h_min hT hT\u2081 hT\u2082 hT\u2083\n have h132 : T\u2081 + T\u2083 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z132 := construct_good_prelim' h_min h132 hT\u2081 hT\u2083 hT\u2082\n have h213 : T\u2082 + T\u2081 + T\u2083 = 0 := by rw [\u2190 hT]; abel\n have Z213 := construct_good_prelim' h_min h213 hT\u2082 hT\u2081 hT\u2083\n have h231 : T\u2082 + T\u2083 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z231 := construct_good_prelim' h_min h231 hT\u2082 hT\u2083 hT\u2081\n have h312 : T\u2083 + T\u2081 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z312 := construct_good_prelim' h_min h312 hT\u2083 hT\u2081 hT\u2082\n have h321 : T\u2083 + T\u2082 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z321 := construct_good_prelim' h_min h321 hT\u2083 hT\u2082 hT\u2081\n simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321\n linarith\n\n/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\nlemma construct_good_improved'' {\u03a9' : Type*} [MeasurableSpace \u03a9'] (\u03bc : Measure \u03a9')\n [IsProbabilityMeasure \u03bc] {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G}\n (hT : T\u2081 + T\u2082 + T\u2083 = 0) (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083) :\n k \u2264 I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc] + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]))\n := by\n let M : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n exact construct_good_improved' h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend aux\n\nopen scoped BigOperators\n\n/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\nlemma averaged_construct_good : k \u2264 (I[U : V | S] + I[V : W | S] + I[W : U | S])\n + (p.\u03b7 / 6) * (((d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]))\n + ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])))\n := by\n have hS : Measurable S := ((hX\u2081.add' hX\u2082).add' hX\u2081').add' hX\u2082'\n have hU : Measurable U := hX\u2081.add' hX\u2082\n have hV : Measurable V := hX\u2081'.add hX\u2082\n have hW : Measurable W := hX\u2081'.add hX\u2081\n have hUVW : U + V + W = 0 := sum_uvw_eq_zero X\u2081 X\u2082 X\u2081'\n have hz (a : \u211d) : a = \u2211 z, (\u2119 (S \u207b\u00b9' {z})).toReal * a := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u2119 hS, one_mul]\n rw [hz k, hz (d[p.X\u2080\u2081 # X\u2081]), hz (d[p.X\u2080\u2082 # X\u2082])]\n simp only [condMutualInfo_eq_sum' hS, \u2190 Finset.sum_add_distrib, \u2190 mul_add,\n condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, \u2190 Finset.sum_sub_distrib, \u2190 mul_sub, Finset.mul_sum,\n \u2190 mul_assoc (p.\u03b7/6), mul_comm (p.\u03b7/6), mul_assoc _ _ (p.\u03b7/6)]\n rw [Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_le_sum (fun i _hi \u21a6 ?_)\n rcases eq_or_ne (\u2119 (S \u207b\u00b9' {i})) 0 with h'i|h'i\n \u00b7 simp [h'i]\n rw [mul_assoc, \u2190 mul_add]\n gcongr\n have : IsProbabilityMeasure (\u2119[|S \u207b\u00b9' {i}]) := cond_isProbabilityMeasure \u2119 h'i\n linarith [construct_good_improved'' h_min (\u2119[|S \u207b\u00b9' {i}]) hUVW hU hV hW]\n\nvariable (p)\n\n", "theoremStatement": "lemma dist_diff_bound_1 :\n (d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n \u2264 (16 * k + 6 * d[X\u2081 # X\u2081] + 2 * d[X\u2082 # X\u2082]) / 4 + (H[X\u2081 + X\u2081'] - H[X\u2082 + X\u2082']) / 4\n + (H[X\u2082 | X\u2082 + X\u2082'] - H[X\u2081 | X\u2081 + X\u2081']) / 4 ", "theoremName": "dist_diff_bound_1", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "0cfc7a4b18a4b56c504490c850af762db44e34cf", "date": "2023-12-14"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 454, "tokenPositionInFile": 23537, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 28, 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"Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have I1 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2082' X\u2081' hX\u2081 hX\u2082 hX\u2082' hX\u2081' h_indep.reindex_four_abcd\n have I2 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep.reindex_four_abdc\n have I3 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2082 X\u2081' hX\u2081 hX\u2082' hX\u2082 hX\u2081' h_indep.reindex_four_acbd\n have I4 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2081' X\u2082 hX\u2081 hX\u2082' hX\u2081' hX\u2082 h_indep.reindex_four_acdb\n have I5 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082 X\u2082' hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_adbc\n have I6 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082' X\u2082 hX\u2081 hX\u2081' hX\u2082' hX\u2082 h_indep.reindex_four_adcb\n have C1 : U + X\u2082' + X\u2081' = S := by abel\n have C2 : W + X\u2082 + X\u2082' = S := by abel\n have C3 : X\u2081 + X\u2082' + X\u2082 + X\u2081' = S := by abel\n have C4 : X\u2081 + X\u2082' + X\u2081' + X\u2082 = S := by abel\n have C5 : W + X\u2082' + X\u2082 = S := by abel\n have C7 : X\u2082 + X\u2081' = V := by abel\n have C8 : X\u2081 + X\u2081' = W := by abel\n have C9 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082.symm\n have C10 : d[X\u2082 # X\u2081'] = d[X\u2081' # X\u2082] := rdist_symm\n have C11 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081.symm\n have C12 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq h\u2082.symm\n have C13 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] := (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082.symm\n have C14 : d[X\u2081' # X\u2082] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C15 : H[X\u2081' + X\u2082'] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C16 : H[X\u2082'] = H[X\u2082] := h\u2082.symm.entropy_eq\n have C17 : H[X\u2081'] = H[X\u2081] := h\u2081.symm.entropy_eq\n have C18 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C19 : H[X\u2082' + X\u2081'] = H[U] := by rw [add_comm]; exact C15\n have C20 : d[X\u2082' # X\u2082] = d[X\u2082 # X\u2082] := h\u2082.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C21 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C22 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C23 : X\u2082' + X\u2082 = X\u2082 + X\u2082' := by abel\n have C24 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C25 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C27 : H[X\u2082' | X\u2082 + X\u2082'] = H[X\u2082 | X\u2082 + X\u2082'] := by\n conv_lhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2082'.add' hX\u2082) hX\u2082 (hX\u2082.add' hX\u2082')\n have I : IdentDistrib (\u27e8X\u2082', X\u2082\u27e9) (\u27e8X\u2082, X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2082.symm h\u2082\n (h_indep.indepFun (show 2 \u2260 1 by decide)) (h_indep.indepFun (show 1 \u2260 2 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C28 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C29 : H[X\u2081' | V] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082) hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C30 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n simp only [C1, C2, C3, C4, C5, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30] at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]", "proofType": "tactic", "proofLengthLines": 88, "proofLengthTokens": 5479}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\nopen ElementaryAddCommGroup\n\n/-- In fact $k$ is at most\n $$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\nlemma construct_good_improved' :\n k \u2264 \u03b4 + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 # T\u2081 | T\u2082] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2082 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2082 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2083 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2083 | T\u2082] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2081 | T\u2082] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2081 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2083 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2083 | T\u2082] - d[p.X\u2080\u2082 # X\u2082])) := by\n have I1 : I[T\u2082 : T\u2081] = I[T\u2081 : T\u2082] := mutualInfo_comm hT\u2082 hT\u2081 _\n have I2 : I[T\u2083 : T\u2081] = I[T\u2081 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2081 _\n have I3 : I[T\u2083 : T\u2082] = I[T\u2082 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2082 _\n have Z123 := construct_good_prelim' h_min hT hT\u2081 hT\u2082 hT\u2083\n have h132 : T\u2081 + T\u2083 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z132 := construct_good_prelim' h_min h132 hT\u2081 hT\u2083 hT\u2082\n have h213 : T\u2082 + T\u2081 + T\u2083 = 0 := by rw [\u2190 hT]; abel\n have Z213 := construct_good_prelim' h_min h213 hT\u2082 hT\u2081 hT\u2083\n have h231 : T\u2082 + T\u2083 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z231 := construct_good_prelim' h_min h231 hT\u2082 hT\u2083 hT\u2081\n have h312 : T\u2083 + T\u2081 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z312 := construct_good_prelim' h_min h312 hT\u2083 hT\u2081 hT\u2082\n have h321 : T\u2083 + T\u2082 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z321 := construct_good_prelim' h_min h321 hT\u2083 hT\u2082 hT\u2081\n simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321\n linarith\n\n/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\nlemma construct_good_improved'' {\u03a9' : Type*} [MeasurableSpace \u03a9'] (\u03bc : Measure \u03a9')\n [IsProbabilityMeasure \u03bc] {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G}\n (hT : T\u2081 + T\u2082 + T\u2083 = 0) (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083) :\n k \u2264 I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc] + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]))\n := by\n let M : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n exact construct_good_improved' h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend aux\n\nopen scoped BigOperators\n\n/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\nlemma averaged_construct_good : k \u2264 (I[U : V | S] + I[V : W | S] + I[W : U | S])\n + (p.\u03b7 / 6) * (((d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]))\n + ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])))\n := by\n have hS : Measurable S := ((hX\u2081.add' hX\u2082).add' hX\u2081').add' hX\u2082'\n have hU : Measurable U := hX\u2081.add' hX\u2082\n have hV : Measurable V := hX\u2081'.add hX\u2082\n have hW : Measurable W := hX\u2081'.add hX\u2081\n have hUVW : U + V + W = 0 := sum_uvw_eq_zero X\u2081 X\u2082 X\u2081'\n have hz (a : \u211d) : a = \u2211 z, (\u2119 (S \u207b\u00b9' {z})).toReal * a := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u2119 hS, one_mul]\n rw [hz k, hz (d[p.X\u2080\u2081 # X\u2081]), hz (d[p.X\u2080\u2082 # X\u2082])]\n simp only [condMutualInfo_eq_sum' hS, \u2190 Finset.sum_add_distrib, \u2190 mul_add,\n condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, \u2190 Finset.sum_sub_distrib, \u2190 mul_sub, Finset.mul_sum,\n \u2190 mul_assoc (p.\u03b7/6), mul_comm (p.\u03b7/6), mul_assoc _ _ (p.\u03b7/6)]\n rw [Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_le_sum (fun i _hi \u21a6 ?_)\n rcases eq_or_ne (\u2119 (S \u207b\u00b9' {i})) 0 with h'i|h'i\n \u00b7 simp [h'i]\n rw [mul_assoc, \u2190 mul_add]\n gcongr\n have : IsProbabilityMeasure (\u2119[|S \u207b\u00b9' {i}]) := cond_isProbabilityMeasure \u2119 h'i\n linarith [construct_good_improved'' h_min (\u2119[|S \u207b\u00b9' {i}]) hUVW hU hV hW]\n\nvariable (p)\n\nlemma dist_diff_bound_1 :\n (d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n \u2264 (16 * k + 6 * d[X\u2081 # X\u2081] + 2 * d[X\u2082 # X\u2082]) / 4 + (H[X\u2081 + X\u2081'] - H[X\u2082 + X\u2082']) / 4\n + (H[X\u2082 | X\u2082 + X\u2082'] - H[X\u2081 | X\u2081 + X\u2081']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2082' X\u2081' hX\u2081 hX\u2082 hX\u2082' hX\u2081' h_indep.reindex_four_abcd\n have I2 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep.reindex_four_abdc\n have I3 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2082 X\u2081' hX\u2081 hX\u2082' hX\u2082 hX\u2081' h_indep.reindex_four_acbd\n have I4 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2081' X\u2082 hX\u2081 hX\u2082' hX\u2081' hX\u2082 h_indep.reindex_four_acdb\n have I5 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082 X\u2082' hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_adbc\n have I6 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082' X\u2082 hX\u2081 hX\u2081' hX\u2082' hX\u2082 h_indep.reindex_four_adcb\n have C1 : U + X\u2082' + X\u2081' = S := by abel\n have C2 : W + X\u2082 + X\u2082' = S := by abel\n have C3 : X\u2081 + X\u2082' + X\u2082 + X\u2081' = S := by abel\n have C4 : X\u2081 + X\u2082' + X\u2081' + X\u2082 = S := by abel\n have C5 : W + X\u2082' + X\u2082 = S := by abel\n have C7 : X\u2082 + X\u2081' = V := by abel\n have C8 : X\u2081 + X\u2081' = W := by abel\n have C9 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082.symm\n have C10 : d[X\u2082 # X\u2081'] = d[X\u2081' # X\u2082] := rdist_symm\n have C11 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081.symm\n have C12 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq h\u2082.symm\n have C13 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] := (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082.symm\n have C14 : d[X\u2081' # X\u2082] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C15 : H[X\u2081' + X\u2082'] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C16 : H[X\u2082'] = H[X\u2082] := h\u2082.symm.entropy_eq\n have C17 : H[X\u2081'] = H[X\u2081] := h\u2081.symm.entropy_eq\n have C18 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C19 : H[X\u2082' + X\u2081'] = H[U] := by rw [add_comm]; exact C15\n have C20 : d[X\u2082' # X\u2082] = d[X\u2082 # X\u2082] := h\u2082.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C21 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C22 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C23 : X\u2082' + X\u2082 = X\u2082 + X\u2082' := by abel\n have C24 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C25 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C27 : H[X\u2082' | X\u2082 + X\u2082'] = H[X\u2082 | X\u2082 + X\u2082'] := by\n conv_lhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2082'.add' hX\u2082) hX\u2082 (hX\u2082.add' hX\u2082')\n have I : IdentDistrib (\u27e8X\u2082', X\u2082\u27e9) (\u27e8X\u2082, X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2082.symm h\u2082\n (h_indep.indepFun (show 2 \u2260 1 by decide)) (h_indep.indepFun (show 1 \u2260 2 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C28 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C29 : H[X\u2081' | V] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082) hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C30 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n simp only [C1, C2, C3, C4, C5, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30] at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\n", "theoremStatement": "lemma dist_diff_bound_2 :\n ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]))\n \u2264 (16 * k + 6 * d[X\u2082 # X\u2082] + 2 * d[X\u2081 # X\u2081]) / 4 + (H[X\u2082 + X\u2082'] - H[X\u2081 + X\u2081']) / 4\n + (H[X\u2081 | X\u2081 + X\u2081'] - H[X\u2082 | X\u2082 + X\u2082']) / 4 ", "theoremName": "dist_diff_bound_2", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "0cfc7a4b18a4b56c504490c850af762db44e34cf", "date": "2023-12-14"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 549, "tokenPositionInFile": 29445, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 28, "numPremises": 321, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", 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"Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have I1 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2082' X\u2081' hX\u2082 hX\u2081 hX\u2082' hX\u2081' h_indep.reindex_four_bacd\n have I2 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2081' X\u2082' hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_badc\n have I3 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081 X\u2081' hX\u2082 hX\u2082' hX\u2081 hX\u2081' h_indep.reindex_four_bcad\n have I4 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081' X\u2081 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bcda\n have I5 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2081 X\u2082' hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bdac\n have I6 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2082' X\u2081 hX\u2082 hX\u2081' hX\u2082' hX\u2081 h_indep.reindex_four_bdca\n have C1 : X\u2082 + X\u2081 = X\u2081 + X\u2082 := by abel\n have C2 : X\u2081 + X\u2081' = W := by abel\n have C3 : U + X\u2082' + X\u2081' = S := by abel\n have C4 : X\u2082 + X\u2081' = V := by abel\n have C5 : X\u2082 + X\u2082' + X\u2081 + X\u2081' = S := by abel\n have C6 : X\u2082 + X\u2082' + X\u2081' + X\u2081 = S := by abel\n have C7 : V + X\u2081 + X\u2082' = S := by abel\n have C8 : V + X\u2082' + X\u2081 = S := by abel\n have C9 : d[X\u2082 # X\u2081] = d[X\u2081 # X\u2082] := rdist_symm\n have C10 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n have C11 : d[X\u2082 # X\u2081'] = d[X\u2081 # X\u2082] := by\n rw [rdist_symm]\n exact ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n have C12 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C13 : d[X\u2082' # X\u2081] = d[X\u2081 # X\u2082'] := rdist_symm\n have C14 : d[X\u2081' # X\u2081] = d[X\u2081 # X\u2081'] := rdist_symm\n have C15 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n have C16 : H[X\u2081' + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C17 : H[X\u2082' + X\u2081'] = H[X\u2081 + X\u2082] := by rw [add_comm]; exact C16\n have C18 : H[X\u2081'] = H[X\u2081] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2081.symm\n have C19 : H[X\u2082'] = H[X\u2082] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2082.symm\n have C20 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C21 : H[X\u2081' | W] = H[X\u2081 | W] := by\n conv_rhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2081) hX\u2081 (hX\u2081.add' hX\u2081')\n have I : IdentDistrib (\u27e8X\u2081', X\u2081\u27e9) (\u27e8X\u2081, X\u2081'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2081\n (h_indep.indepFun (show 3 \u2260 0 by decide)) (h_indep.indepFun (show 0 \u2260 3 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C22 : H[X\u2082' | X\u2082' + X\u2081] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C23 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C24 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C25 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C27 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n have C28 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C29 : H[X\u2082' + X\u2081] = H[X\u2081 + X\u2082] := by\n rw [add_comm]\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C30 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2081.symm\n have C31 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable) h\u2082.symm\n simp only [C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C25, C26, C27, C28, C29, C30, C31]\n at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]", "proofType": "tactic", "proofLengthLines": 104, "proofLengthTokens": 6199}} +{"srcContext": "import PFR.FirstEstimate\nimport PFR.Mathlib.Probability.Independence.FourVariables\n\n/-!\n# Second estimate\n\nThe second estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ be independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $d[X_1;X_2] = k$\n* $I_1 := I_1 [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n* $I_2 := I[X_1+X_2 : X_1 + \\tilde X_1 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `second_estimate` : $$ I_2 \\leq 2 \\eta k + \\frac{2 \\eta (2 \\eta k - I_1)}{1 - \\eta}.$$\n-/\n\nopen MeasureTheory ProbabilityTheory ElementaryAddCommGroup\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2081', X\u2082'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- $$ d[X_1+\\tilde X_1; X_2+\\tilde X_2] \\geq k - \\frac{\\eta}{2} ( d[X_1; X_1] + d[X_2;X_2] ).$$\n-/\nlemma rdist_of_sums_ge' : d[X\u2081 + X\u2081' # X\u2082 + X\u2082'] \u2265 k - p.\u03b7 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082]) / 2 := by\n refine' LE.le.ge (LE.le.trans _ (distance_ge_of_min p h_min (hX\u2081.add hX\u2081') (hX\u2082.add hX\u2082')))\n rw [sub_sub, sub_le_sub_iff_left k, \u2190 mul_add,mul_div_assoc]\n refine' mul_le_mul_of_nonneg_left _ (by linarith [p.h\u03b7])\n have h\u2081' := condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2081' (h_indep.indepFun (show 0 \u2260 2 by decide))\n have h\u2082' := condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2082' (h_indep.indepFun (show 1 \u2260 3 by decide))\n rw [h\u2081.entropy_eq, add_sub_cancel_right, \u2190 (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081] at h\u2081'\n rw [h\u2082.entropy_eq, add_sub_cancel_right, \u2190 (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082] at h\u2082'\n linarith\n\nlemma second_estimate_aux :\n d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082] \u2264 2 * (k + (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7)) := by\n have hX\u2081_indep : IndepFun X\u2081 X\u2081' (\u03bc := \u2119) := h_indep.indepFun (show 0 \u2260 2 by decide)\n have hX\u2082_indep : IndepFun X\u2082 X\u2082' (\u03bc := \u2119) := h_indep.indepFun (show 1 \u2260 3 by decide)\n have hX_indep : IndepFun (X\u2081 + X\u2081') (X\u2082 + X\u2082') := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption) 0 2 1 3\n (by decide) (by decide) (by decide) (by decide)\n have h : d[X\u2081 + X\u2081' # X\u2082+ X\u2082'] \u2264 (2 + p.\u03b7) * k - (d[X\u2081# X\u2081] + d[X\u2082 # X\u2082]) / 2 - I\u2081 := by\n have h := hX_indep.rdist_eq (hX\u2081.add hX\u2081') (hX\u2082.add hX\u2082')\n rw [sub_eq_add (X\u2081 + X\u2081') (X\u2082 + X\u2082'), \u2190 sub_eq_add X\u2081 X\u2081', \u2190 sub_eq_add X\u2082 X\u2082',\n sub_eq_iff_eq_add.mp (sub_eq_iff_eq_add.mp (hX\u2081_indep.rdist_eq hX\u2081 hX\u2081').symm),\n sub_eq_iff_eq_add.mp (sub_eq_iff_eq_add.mp (hX\u2082_indep.rdist_eq hX\u2082 hX\u2082').symm),\n \u2190 h\u2081.entropy_eq, \u2190 h\u2082.entropy_eq, add_assoc, add_assoc, add_halves', add_halves',\n \u2190 (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081,\n \u2190 (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082,\n sub_eq_add X\u2081 X\u2081', sub_eq_add X\u2082 X\u2082', \u2190 add_assoc, add_right_comm _ X\u2081'] at h\n have h_indep' : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'] :=\n by exact h_indep.reindex_four_abdc\n have h' := ent_ofsum_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep' h_min\n convert (h.symm \u25b8 (sub_le_sub_right (sub_le_sub_right h' _) _)) using 1; ring\n have h' := (rdist_of_sums_ge' p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min).le.trans h\n rw [\u2190 div_le_iff' two_pos, \u2190 sub_le_iff_le_add', le_div_iff (by linarith [p.h\u03b7'])]\n linarith\n\n", "theoremStatement": "/-- $$ I_2 \\leq 2 \\eta k + \\frac{2 \\eta (2 \\eta k - I_1)}{1 - \\eta}.$$ -/\nlemma second_estimate : I\u2082 \u2264 2 * p.\u03b7 * k + (2 * p.\u03b7 * (2 * p.\u03b7 * k - I\u2081)) / (1 - p.\u03b7) ", "theoremName": 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"Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have hX\u2081_indep : IndepFun X\u2081 X\u2081' (\u03bc := \u2119) := h_indep.indepFun (show 0 \u2260 2 by decide)\n have hX\u2082_indep : IndepFun X\u2082 X\u2082' (\u03bc := \u2119) := h_indep.indepFun (show 1 \u2260 3 by decide)\n let Y : Fin 4 \u2192 \u03a9 \u2192 G := ![X\u2082, X\u2081, X\u2082', X\u2081']\n have hY : \u2200 i, Measurable (Y i) := fun i => by fin_cases i <;> assumption\n have hY_indep : iIndepFun (fun _ => hG) Y := by exact h_indep.reindex_four_badc\n have h := sum_of_rdist_eq_char_2 Y hY_indep hY\n rw [show Y 0 = X\u2082 by rfl, show Y 1 = X\u2081 by rfl, show Y 2 = X\u2082' by rfl, show Y 3 = X\u2081' by rfl] at h\n rw [\u2190 h\u2082.rdist_eq h\u2081, rdist_symm, rdist_symm (X := X\u2082 + X\u2082'),\n condRuzsaDist_symm (Z := X\u2082 + X\u2082') (W := X\u2081 + X\u2081') (hX\u2082.add hX\u2082') (hX\u2081.add hX\u2081'),\n \u2190 two_mul] at h\n replace h : 2 * k = d[X\u2081 + X\u2081' # X\u2082 + X\u2082'] + d[X\u2081 | X\u2081 + X\u2081' # X\u2082 | X\u2082 + X\u2082'] + I[X\u2081 + X\u2082 : X\u2081 + X\u2081'|X\u2081 + X\u2082 + X\u2081' + X\u2082'] := by\n convert h using 3 <;> abel\n have h' := condRuzsaDistance_ge_of_min p h_min hX\u2081 hX\u2082 (X\u2081 + X\u2081') (X\u2082 + X\u2082') (hX\u2081.add hX\u2081') (hX\u2082.add hX\u2082')\n have h\u2081' := condRuzsaDist_diff_le''' \u2119 p.hmeas1 hX\u2081 hX\u2081' hX\u2081_indep\n have h\u2082' := condRuzsaDist_diff_le''' \u2119 p.hmeas2 hX\u2082 hX\u2082' hX\u2082_indep\n rw [h\u2081.entropy_eq, add_sub_cancel_right, \u2190 (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081] at h\u2081'\n rw [h\u2082.entropy_eq, add_sub_cancel_right, \u2190 (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082] at h\u2082'\n have h'' : I\u2082 \u2264 p.\u03b7 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082]) := by\n simp_rw [\u2190 add_comm X\u2081 X\u2081']\n have h\u2081'' := mul_le_mul_of_nonneg_left h\u2081' (show 0 \u2264 p.\u03b7 by linarith [p.h\u03b7])\n have h\u2082'' := mul_le_mul_of_nonneg_left h\u2082' (show 0 \u2264 p.\u03b7 by linarith [p.h\u03b7])\n have := rdist_of_sums_ge' p _ _ _ _ hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n linarith\n nth_rewrite 1 [mul_div_assoc, \u2190 mul_add, mul_assoc, mul_left_comm]\n refine' h''.trans (mul_le_mul_of_nonneg_left _ (show 0 \u2264 p.\u03b7 by linarith [p.h\u03b7]))\n exact second_estimate_aux p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min", "proofType": "tactic", "proofLengthLines": 26, "proofLengthTokens": 1892}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n", "theoremStatement": "/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] ", "theoremName": "construct_good_prelim'", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 319, "tokenPositionInFile": 15804, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, 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"Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "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", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "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", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4", "proofType": "tactic", "proofLengthLines": 46, "proofLengthTokens": 2704}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n", "theoremStatement": "/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] ", "theoremName": "ent_of_proj_le", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "5b414088d10537ec7660995df68362d19a6f8ede", "date": "2023-12-18"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 247, "tokenPositionInFile": 12399, "theoremPositionInFile": 20}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 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"Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]", "proofType": "tactic", "proofLengthLines": 75, "proofLengthTokens": 4306}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\nopen ElementaryAddCommGroup\n\n/-- In fact $k$ is at most\n $$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\nlemma construct_good_improved' :\n k \u2264 \u03b4 + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 # T\u2081 | T\u2082] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2082 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2082 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2083 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2083 | T\u2082] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2081 | T\u2082] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2081 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2083 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2083 | T\u2082] - d[p.X\u2080\u2082 # X\u2082])) := by\n have I1 : I[T\u2082 : T\u2081] = I[T\u2081 : T\u2082] := mutualInfo_comm hT\u2082 hT\u2081 _\n have I2 : I[T\u2083 : T\u2081] = I[T\u2081 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2081 _\n have I3 : I[T\u2083 : T\u2082] = I[T\u2082 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2082 _\n have Z123 := construct_good_prelim' h_min hT hT\u2081 hT\u2082 hT\u2083\n have h132 : T\u2081 + T\u2083 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z132 := construct_good_prelim' h_min h132 hT\u2081 hT\u2083 hT\u2082\n have h213 : T\u2082 + T\u2081 + T\u2083 = 0 := by rw [\u2190 hT]; abel\n have Z213 := construct_good_prelim' h_min h213 hT\u2082 hT\u2081 hT\u2083\n have h231 : T\u2082 + T\u2083 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z231 := construct_good_prelim' h_min h231 hT\u2082 hT\u2083 hT\u2081\n have h312 : T\u2083 + T\u2081 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z312 := construct_good_prelim' h_min h312 hT\u2083 hT\u2081 hT\u2082\n have h321 : T\u2083 + T\u2082 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z321 := construct_good_prelim' h_min h321 hT\u2083 hT\u2082 hT\u2081\n simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321\n linarith\n\n/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\nlemma construct_good_improved'' {\u03a9' : Type*} [MeasurableSpace \u03a9'] (\u03bc : Measure \u03a9')\n [IsProbabilityMeasure \u03bc] {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G}\n (hT : T\u2081 + T\u2082 + T\u2083 = 0) (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083) :\n k \u2264 I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc] + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]))\n := by\n let M : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n exact construct_good_improved' h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend aux\n\nopen scoped BigOperators\n\n/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\nlemma averaged_construct_good : k \u2264 (I[U : V | S] + I[V : W | S] + I[W : U | S])\n + (p.\u03b7 / 6) * (((d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]))\n + ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])))\n := by\n have hS : Measurable S := ((hX\u2081.add' hX\u2082).add' hX\u2081').add' hX\u2082'\n have hU : Measurable U := hX\u2081.add' hX\u2082\n have hV : Measurable V := hX\u2081'.add hX\u2082\n have hW : Measurable W := hX\u2081'.add hX\u2081\n have hUVW : U + V + W = 0 := sum_uvw_eq_zero X\u2081 X\u2082 X\u2081'\n have hz (a : \u211d) : a = \u2211 z, (\u2119 (S \u207b\u00b9' {z})).toReal * a := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u2119 hS, one_mul]\n rw [hz k, hz (d[p.X\u2080\u2081 # X\u2081]), hz (d[p.X\u2080\u2082 # X\u2082])]\n simp only [condMutualInfo_eq_sum' hS, \u2190 Finset.sum_add_distrib, \u2190 mul_add,\n condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, \u2190 Finset.sum_sub_distrib, \u2190 mul_sub, Finset.mul_sum,\n \u2190 mul_assoc (p.\u03b7/6), mul_comm (p.\u03b7/6), mul_assoc _ _ (p.\u03b7/6)]\n rw [Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_le_sum (fun i _hi \u21a6 ?_)\n rcases eq_or_ne (\u2119 (S \u207b\u00b9' {i})) 0 with h'i|h'i\n \u00b7 simp [h'i]\n rw [mul_assoc, \u2190 mul_add]\n gcongr\n have : IsProbabilityMeasure (\u2119[|S \u207b\u00b9' {i}]) := cond_isProbabilityMeasure \u2119 h'i\n linarith [construct_good_improved'' h_min (\u2119[|S \u207b\u00b9' {i}]) hUVW hU hV hW]\n\nvariable (p)\n\nlemma dist_diff_bound_1 :\n (d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n \u2264 (16 * k + 6 * d[X\u2081 # X\u2081] + 2 * d[X\u2082 # X\u2082]) / 4 + (H[X\u2081 + X\u2081'] - H[X\u2082 + X\u2082']) / 4\n + (H[X\u2082 | X\u2082 + X\u2082'] - H[X\u2081 | X\u2081 + X\u2081']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2082' X\u2081' hX\u2081 hX\u2082 hX\u2082' hX\u2081' h_indep.reindex_four_abcd\n have I2 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep.reindex_four_abdc\n have I3 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2082 X\u2081' hX\u2081 hX\u2082' hX\u2082 hX\u2081' h_indep.reindex_four_acbd\n have I4 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2081' X\u2082 hX\u2081 hX\u2082' hX\u2081' hX\u2082 h_indep.reindex_four_acdb\n have I5 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082 X\u2082' hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_adbc\n have I6 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082' X\u2082 hX\u2081 hX\u2081' hX\u2082' hX\u2082 h_indep.reindex_four_adcb\n have C1 : U + X\u2082' + X\u2081' = S := by abel\n have C2 : W + X\u2082 + X\u2082' = S := by abel\n have C3 : X\u2081 + X\u2082' + X\u2082 + X\u2081' = S := by abel\n have C4 : X\u2081 + X\u2082' + X\u2081' + X\u2082 = S := by abel\n have C5 : W + X\u2082' + X\u2082 = S := by abel\n have C7 : X\u2082 + X\u2081' = V := by abel\n have C8 : X\u2081 + X\u2081' = W := by abel\n have C9 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082.symm\n have C10 : d[X\u2082 # X\u2081'] = d[X\u2081' # X\u2082] := rdist_symm\n have C11 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081.symm\n have C12 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq h\u2082.symm\n have C13 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] := (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082.symm\n have C14 : d[X\u2081' # X\u2082] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C15 : H[X\u2081' + X\u2082'] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C16 : H[X\u2082'] = H[X\u2082] := h\u2082.symm.entropy_eq\n have C17 : H[X\u2081'] = H[X\u2081] := h\u2081.symm.entropy_eq\n have C18 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C19 : H[X\u2082' + X\u2081'] = H[U] := by rw [add_comm]; exact C15\n have C20 : d[X\u2082' # X\u2082] = d[X\u2082 # X\u2082] := h\u2082.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C21 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C22 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C23 : X\u2082' + X\u2082 = X\u2082 + X\u2082' := by abel\n have C24 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C25 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C27 : H[X\u2082' | X\u2082 + X\u2082'] = H[X\u2082 | X\u2082 + X\u2082'] := by\n conv_lhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2082'.add' hX\u2082) hX\u2082 (hX\u2082.add' hX\u2082')\n have I : IdentDistrib (\u27e8X\u2082', X\u2082\u27e9) (\u27e8X\u2082, X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2082.symm h\u2082\n (h_indep.indepFun (show 2 \u2260 1 by decide)) (h_indep.indepFun (show 1 \u2260 2 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C28 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C29 : H[X\u2081' | V] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082) hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C30 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n simp only [C1, C2, C3, C4, C5, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30] at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma dist_diff_bound_2 :\n ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]))\n \u2264 (16 * k + 6 * d[X\u2082 # X\u2082] + 2 * d[X\u2081 # X\u2081]) / 4 + (H[X\u2082 + X\u2082'] - H[X\u2081 + X\u2081']) / 4\n + (H[X\u2081 | X\u2081 + X\u2081'] - H[X\u2082 | X\u2082 + X\u2082']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2082' X\u2081' hX\u2082 hX\u2081 hX\u2082' hX\u2081' h_indep.reindex_four_bacd\n have I2 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2081' X\u2082' hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_badc\n have I3 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081 X\u2081' hX\u2082 hX\u2082' hX\u2081 hX\u2081' h_indep.reindex_four_bcad\n have I4 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081' X\u2081 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bcda\n have I5 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2081 X\u2082' hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bdac\n have I6 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2082' X\u2081 hX\u2082 hX\u2081' hX\u2082' hX\u2081 h_indep.reindex_four_bdca\n have C1 : X\u2082 + X\u2081 = X\u2081 + X\u2082 := by abel\n have C2 : X\u2081 + X\u2081' = W := by abel\n have C3 : U + X\u2082' + X\u2081' = S := by abel\n have C4 : X\u2082 + X\u2081' = V := by abel\n have C5 : X\u2082 + X\u2082' + X\u2081 + X\u2081' = S := by abel\n have C6 : X\u2082 + X\u2082' + X\u2081' + X\u2081 = S := by abel\n have C7 : V + X\u2081 + X\u2082' = S := by abel\n have C8 : V + X\u2082' + X\u2081 = S := by abel\n have C9 : d[X\u2082 # X\u2081] = d[X\u2081 # X\u2082] := rdist_symm\n have C10 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n have C11 : d[X\u2082 # X\u2081'] = d[X\u2081 # X\u2082] := by\n rw [rdist_symm]\n exact ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n have C12 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C13 : d[X\u2082' # X\u2081] = d[X\u2081 # X\u2082'] := rdist_symm\n have C14 : d[X\u2081' # X\u2081] = d[X\u2081 # X\u2081'] := rdist_symm\n have C15 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n have C16 : H[X\u2081' + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C17 : H[X\u2082' + X\u2081'] = H[X\u2081 + X\u2082] := by rw [add_comm]; exact C16\n have C18 : H[X\u2081'] = H[X\u2081] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2081.symm\n have C19 : H[X\u2082'] = H[X\u2082] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2082.symm\n have C20 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C21 : H[X\u2081' | W] = H[X\u2081 | W] := by\n conv_rhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2081) hX\u2081 (hX\u2081.add' hX\u2081')\n have I : IdentDistrib (\u27e8X\u2081', X\u2081\u27e9) (\u27e8X\u2081, X\u2081'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2081\n (h_indep.indepFun (show 3 \u2260 0 by decide)) (h_indep.indepFun (show 0 \u2260 3 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C22 : H[X\u2082' | X\u2082' + X\u2081] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C23 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C24 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C25 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C27 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n have C28 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C29 : H[X\u2082' + X\u2081] = H[X\u2081 + X\u2082] := by\n rw [add_comm]\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C30 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2081.symm\n have C31 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable) h\u2082.symm\n simp only [C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C25, C26, C27, C28, C29, C30, C31]\n at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma averaged_final : k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) := by\n apply (averaged_construct_good hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_min).trans\n have : 0 \u2264 p.\u03b7 := p.h\u03b7.le\n have := sum_condMutual_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep.reindex_four_abdc h_min\n gcongr ?_ + (p.\u03b7 / 6) * ?_\n linarith [dist_diff_bound_1 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep,\n dist_diff_bound_2 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep]\n\n/-- Suppose $0 < \\eta < 1/8$. Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$. The proof\nof this lemma uses copies `X\u2081', X\u2082'` already in the context. For a version that does not assume\nthese are given and constructs them instead, use `tau_strictly_decreases'`.\n-/\ntheorem tau_strictly_decreases_aux' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 := by\n have : 0 < p.\u03b7 := p.h\u03b7\n have : k \u2264 8 * p.\u03b7 * k := by calc\n k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) :=\n averaged_final p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n _ \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) +\n p.\u03b7 / 6 * (8 * k + 2 * (2 * (k + (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7)))) := by\n gcongr\n exact second_estimate_aux p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082\n h_indep.reindex_four_abdc h_min\n _ = 8 * p.\u03b7 * k - (1 - 5 * p.\u03b7 - 4/6 * p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7) := by ring\n _ \u2264 8 * p.\u03b7 * k := by\n simp only [tsub_le_iff_right, le_add_iff_nonneg_right]\n apply div_nonneg _ (by linarith)\n apply mul_nonneg (by linarith) _\n linarith [first_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min]\n apply le_antisymm _ (rdist_nonneg hX\u2081 hX\u2082)\n nlinarith\n\ntheorem tau_strictly_decreases' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 := by\n let \u27e8A, mA, \u03bc, Y\u2081, Y\u2082, Y\u2081', Y\u2082', h\u03bc, h_indep, hY\u2081, hY\u2082, hY\u2081', hY\u2082', h_id1, h_id2, h_id1', h_id2'\u27e9\n := independent_copies4_nondep hX\u2081 hX\u2082 hX\u2081 hX\u2082 \u2119 \u2119 \u2119 \u2119\n rw [\u2190 h_id1.rdist_eq h_id2]\n letI : MeasureSpace A := \u27e8\u03bc\u27e9\n have : IsProbabilityMeasure (\u2119 : Measure A) := h\u03bc\n rw [\u2190 h_id1.tau_minimizes p h_id2] at h_min\n exact tau_strictly_decreases_aux' p hY\u2081 hY\u2082 hY\u2081' hY\u2082' (h_id1.trans h_id1'.symm)\n (h_id2.trans h_id2'.symm) h_indep.reindex_four_abdc h_min hp\n\nend MainEstimates\n\nsection EntropicPFR\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nopen scoped Topology\nopen Filter Set\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n [MeasurableSingletonClass G]\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\n", "theoremStatement": "/-- For `p.\u03b7 \u2264 1/8`, there exist \u03c4-minimizers `X\u2081, X\u2082` at zero Rusza distance. For `p.\u03b7 < 1/8`,\nall minimizers are fine, by `tau_strictly_decreases'`. For `p.\u03b7 = 1/8`, we use a limit of\nminimizers for `\u03b7 < 1/8`, which exists by compactness. -/\nlemma tau_minimizer_exists_rdist_eq_zero :\n \u2203 (\u03a9 : Type uG) (m\u03a9 : MeasureSpace \u03a9) (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G),\n Measurable X\u2081 \u2227 Measurable X\u2082 \u2227 IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 tau_minimizes p X\u2081 X\u2082\n \u2227 d[X\u2081 # X\u2082] = 0 ", "theoremName": "tau_minimizer_exists_rdist_eq_zero", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "d7b6033604422555584766a86b127ec46025a905", "date": "2023-12-15"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 720, "tokenPositionInFile": 38843, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, 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"Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "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", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "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", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n -- let `u\u2099` be a sequence converging from below to `\u03b7`. In particular, `u\u2099 < 1/8`.\n obtain \u27e8u, -, u_mem, u_lim\u27e9 :\n \u2203 u, StrictMono u \u2227 (\u2200 (n : \u2115), u n \u2208 Set.Ioo 0 p.\u03b7) \u2227 Tendsto u atTop (\ud835\udcdd p.\u03b7) :=\n exists_seq_strictMono_tendsto' p.h\u03b7\n -- For each `n`, consider a minimizer associated to `\u03b7 = u\u2099`.\n let q : \u2115 \u2192 refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G := fun n \u21a6\n \u27e8p.X\u2080\u2081, p.X\u2080\u2082, p.hmeas1, p.hmeas2, u n, (u_mem n).1, by linarith [(u_mem n).2, p.h\u03b7']\u27e9\n have : \u2200 n, \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | q n] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | q n] :=\n fun n \u21a6 tau_min_exists_measure (q n)\n choose \u03bc \u03bc1_prob \u03bc2_prob h\u03bc using this\n -- The minimizer associated to `u\u2099` is at zero Rusza distance of itself, by\n -- lemma `tau_strictly_decreases'`.\n have I n : d[id ; (\u03bc n).1 # id ; (\u03bc n).2] = 0 := by\n let M : MeasureSpace (G \u00d7 G) := \u27e8(\u03bc n).1.prod (\u03bc n).2\u27e9\n have : IsProbabilityMeasure ((\u03bc n).1.prod (\u03bc n).2) := by infer_instance\n have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; (\u03bc n).1 # id ; (\u03bc n).2] :=\n IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id\n rw [\u2190 this]\n apply tau_strictly_decreases' (q n) measurable_fst measurable_snd ?_\n (by linarith [(u_mem n).2, p.h\u03b7'])\n intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | q n] = \u03c4[id ; (\u03bc n).1 # id ; (\u03bc n).2 | q n] :=\n ProbabilityTheory.IdentDistrib.tau_eq (q n) IdentDistrib.fst_id IdentDistrib.snd_id\n rw [A]\n exact h\u03bc n _ _ h\u2081 h\u2082\n -- extract a converging subsequence of the sequence of minimizers, seen as pairs of probability\n -- measures on `G` (which is a compact space).\n let \u03bc' : \u2115 \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun n \u21a6 (\u27e8(\u03bc n).1, \u03bc1_prob n\u27e9, \u27e8(\u03bc n).2, \u03bc2_prob n\u27e9)\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G)\n have : DiscreteTopology G := \u27e8rfl\u27e9\n -- The limiting pair of measures will be the desired minimizer.\n rcases IsCompact.tendsto_subseq (x := \u03bc') isCompact_univ (fun n \u21a6 mem_univ _)\n with \u27e8\u03bd, -, \u03c6, \u03c6mono, h\u03bd\u27e9\n have \u03c6lim : Tendsto \u03c6 atTop atTop := \u03c6mono.tendsto_atTop\n let M : MeasureSpace (G \u00d7 G) := \u27e8(\u03bd.1 : Measure G).prod \u03bd.2\u27e9\n have P : IsProbabilityMeasure ((\u03bd.1 : Measure G).prod (\u03bd.2 : Measure G)) := by infer_instance\n refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_, ?_\u27e9\n -- check that it is indeed a minimizer, as a limit of minimizers.\n \u00b7 intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bd.1 # id ; \u03bd.2 | p] :=\n ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n rw [A]\n have L1 : Tendsto (fun n \u21a6 \u03c4[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2 | q (\u03c6 n)]) atTop\n (\ud835\udcdd (\u03c4[id ; \u03bd.1 # id ; \u03bd.2 | p])) := by\n apply Tendsto.add (Tendsto.add ?_ (Tendsto.mul (u_lim.comp \u03c6lim) ?_))\n (Tendsto.mul (u_lim.comp \u03c6lim) ?_)\n \u00b7 apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) h\u03bd\n \u00b7 have : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n apply Tendsto.comp (this.tendsto _) h\u03bd\n \u00b7 have : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n apply Tendsto.comp (this.tendsto _) h\u03bd\n have L2 : Tendsto (fun n \u21a6 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | q (\u03c6 n)]) atTop\n (\ud835\udcdd (\u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p])) :=\n Tendsto.add (Tendsto.add tendsto_const_nhds (Tendsto.mul (u_lim.comp \u03c6lim)\n tendsto_const_nhds)) (Tendsto.mul (u_lim.comp \u03c6lim) tendsto_const_nhds)\n exact le_of_tendsto_of_tendsto' L1 L2 (fun n \u21a6 h\u03bc (\u03c6 n) _ _ h\u2081 h\u2082)\n -- check that it has zero Rusza distance, as a limit of a sequence at zero Rusza distance.\n \u00b7 have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; \u03bd.1 # id ; \u03bd.2] :=\n IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id\n rw [this]\n have L1 : Tendsto (fun n \u21a6 d[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2]) atTop\n (\ud835\udcdd (d[id ; \u03bd.1 # id ; (\u03bd.2 : Measure G)])) := by\n apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) h\u03bd\n have L2 : Tendsto (fun n \u21a6 d[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2]) atTop (\ud835\udcdd 0) := by simp [I]\n exact tendsto_nhds_unique L1 L2", "proofType": "tactic", "proofLengthLines": 73, "proofLengthTokens": 4553}} +{"srcContext": "import PFR.FirstEstimate\nimport PFR.SecondEstimate\nimport PFR.ForMathlib.Elementary\n\n/-!\n# Endgame\n\nThe endgame on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ be independent random variables, with $X_1,\\tilde X_1$ copies\n of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $d[X_1;X_2] = k$\n* $U := X_1 + X_2$\n* $V := \\tilde X_1 + X_2$\n* $W := X_1 + \\tilde X_1$\n* $S := X_1 + X_2 + \\tilde X_1 + \\tilde X_2$.\n* $I_1 := I[U : V | S]$\n* $I_2 := I[U : W | S]$\n* $I_3 := I[V : W | S]$ (not explicitly defined in Lean)\n\n# Main results:\n\n* `sum_condMutual_le` : An upper bound on the total conditional mutual information $I_1+I_2+I_3$.\n* `sum_dist_diff_le`: A sum of the \"costs\" of $U$, $V$, $W$.\n* `construct_good`: A construction of two random variables with small Ruzsa distance between them\n given some random variables with control on total cost, as well as total mutual information.\n-/\n\nopen MeasureTheory ProbabilityTheory\nopen scoped BigOperators\n\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [m\u03a9 : MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2081', X\u2082'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[U : V | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `V = X\u2081' + X\u2082`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[U : V | S]\n\n/-- `I\u2082 := I[U : W | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `W = X\u2081' + X\u2081`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[U : W | S]\n\n--(Mantas) this times out in the proof below\nprivate lemma hmeas2 :\n Measurable fun p : Fin 4 \u2192 G => ((p 0 + p 1, p 0 + p 2), p 0 + p 1 + p 2 + p 3) := by\n apply Measurable.prod\n \u00b7 apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n\n/-- The quantity $I_3 = I[V:W|S]$ is equal to $I_2$. -/\nlemma I\u2083_eq : I[V : W | S] = I\u2082 := by\n have h_indep2 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081, X\u2082'] := by\n exact h_indep.reindex_four_cbad\n have hident : IdentDistrib (fun a (i : Fin 4) => ![X\u2081, X\u2082, X\u2081', X\u2082'] i a)\n (fun a (j : Fin 4) => ![X\u2081', X\u2082, X\u2081, X\u2082'] j a) := by\n exact { aemeasurable_fst := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n aemeasurable_snd := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n map_eq := by\n rw [\u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081, X\u2082, X\u2081', X\u2082'])\n (Fin.cases hX\u2081 <| Fin.cases hX\u2082 <| Fin.cases hX\u2081' <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep,\n \u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081', X\u2082, X\u2081, X\u2082'])\n (Fin.cases hX\u2081' <| Fin.cases hX\u2082 <| Fin.cases hX\u2081 <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep2]\n congr\n ext i\n fin_cases i\n all_goals simp [h\u2081.map_eq] }\n have hmeas1 : Measurable (fun p : Fin 4 \u2192 G => (p 0 + p 1, p 0 + p 1 + p 2 + p 3)) := by\n simp_all only [Matrix.cons_val', Matrix.empty_val', Matrix.cons_val_fin_one]\n apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n have hUVS : IdentDistrib (prod U S) (prod V S) := by\n convert (IdentDistrib.comp hident hmeas1); simp; abel\n have hUVWS : IdentDistrib (prod (prod U W) S) (prod (prod V W) S) := by\n convert (IdentDistrib.comp hident hmeas2) <;> simp <;> abel\n have hU : Measurable U := Measurable.add hX\u2081 hX\u2082\n have hV : Measurable V := Measurable.add hX\u2081' hX\u2082\n have hW : Measurable W := Measurable.add hX\u2081' hX\u2081\n have hS : Measurable S := by measurability\n rw [condMutualInfo_eq hV hW hS, condMutualInfo_eq hU hW hS, chain_rule'' \u2119 hU hS,\n chain_rule'' \u2119 hV hS, chain_rule'' \u2119 hW hS, chain_rule'' \u2119 _ hS, chain_rule'' \u2119 _ hS,\n IdentDistrib.entropy_eq hUVS, IdentDistrib.entropy_eq hUVWS]\n \u00b7 exact Measurable.prod (by exact hU) (by exact hW)\n \u00b7 exact Measurable.prod (by exact hV) (by exact hW)\n\n\n/--\n$$ I(U : V | S) + I(V : W | S) + I(W : U | S) $$\nis less than or equal to\n$$ 6 \\eta k - \\frac{1 - 5 \\eta}{1-\\eta} (2 \\eta k - I_1).$$\n-/\nlemma sum_condMutual_le :\n I[U : V | S] + I[V : W | S] + I[W : U | S]\n \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) := by\n have : I[W:U|S] = I\u2082 := by\n rw [condMutualInfo_comm]\n \u00b7 exact Measurable.add' hX\u2081' hX\u2081\n \u00b7 exact Measurable.add' hX\u2081 hX\u2082\n rw [I\u2083_eq, this]\n have h\u2082 := second_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n have h := add_le_add (add_le_add_left h\u2082 I\u2081) h\u2082\n convert h using 1\n have : 1 - p.\u03b7 > 0 := by linarith [p.h\u03b7']\n field_simp [this]\n ring\n all_goals { simpa }\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" => d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nlemma hU : H[U] = H[X\u2081' + X\u2082'] :=\n IdentDistrib.entropy_eq (ProbabilityTheory.IdentDistrib.add h\u2081 h\u2082\n (iIndepFun.indepFun h_indep (show (0 : Fin 4) \u2260 1 by norm_cast))\n (iIndepFun.indepFun h_indep (show (2 : Fin 4) \u2260 3 by norm_cast)))\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition1 : iIndepFun (fun _ \u21a6 hG) ![X\u2081, X\u2082, X\u2081' + X\u2082'] :=\n h_indep.apply_two_last hX\u2081 hX\u2082 hX\u2081' hX\u2082' measurable_add\n\nlemma hV : H[V] = H[X\u2081 + X\u2082'] :=\nIdentDistrib.entropy_eq (ProbabilityTheory.IdentDistrib.add h\u2081.symm h\u2082\n (iIndepFun.indepFun h_indep (show (2 : Fin 4) \u2260 1 by norm_cast))\n (iIndepFun.indepFun h_indep (show (0 : Fin 4) \u2260 3 by norm_cast)))\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition2 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2081, X\u2081' + X\u2082'] :=\n independenceCondition1 hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_bacd\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition3 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081 + X\u2082'] :=\n independenceCondition1 hX\u2081' hX\u2082 hX\u2081 hX\u2082' h_indep.reindex_four_cbad\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition4 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2081', X\u2081 + X\u2082'] :=\n independenceCondition1 hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bcad\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition5 : iIndepFun (fun _ \u21a6 hG) ![X\u2081, X\u2081', X\u2082 + X\u2082'] :=\n independenceCondition1 hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_acbd\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition6 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2082', X\u2081' + X\u2081] :=\n independenceCondition1 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bdca\n\nset_option maxHeartbeats 400000 in\n", "theoremStatement": "/--\n$$ \\sum_{i=1}^2 \\sum_{A\\in\\{U,V,W\\}} \\big(d[X^0_i;A|S] - d[X^0_i;X_i]\\big)$$\nis less than or equal to\n$$ \\leq (6 - 3\\eta) k + 3(2 \\eta k - I_1).$$\n-/\nlemma sum_dist_diff_le :\n c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 (6 - 3 * p.\u03b7)*k + 3 * (2*p.\u03b7*k - I\u2081) ", "theoremName": "sum_dist_diff_le", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "85f77106b3f388041ddae5a52f3631cf0eb16c44", "date": "2023-11-16"}, "file": "PFR/PFR/Endgame.lean", "module": "PFR.Endgame", "jsonFile": "PFR.Endgame.jsonl", "positionMetadata": {"lineInFile": 208, "tokenPositionInFile": 8132, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 8, "repositoryPremises": true, "numRepositoryPremises": 28, 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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.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let X\u2080\u2081 := p.X\u2080\u2081\n let X\u2080\u2082 := p.X\u2080\u2082\n have ineq1 : d[X\u2080\u2081 # U | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 := by\n have aux1 : H[S] + H[U] - H[X\u2081] - H[X\u2081' + X\u2082'] = H[S] - H[X\u2081] := by\n rw [hU X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep]; ring\n have aux2 : d[X\u2080\u2081 # U | U + (X\u2081' + X\u2082')] - d[X\u2080\u2081 # X\u2081]\n \u2264 (H[U + (X\u2081' + X\u2082')] + H[U] - H[X\u2081] - H[X\u2081' + X\u2082']) / 2 :=\n condRuzsaDist_diff_ofsum_le \u2119 (hX := p.hmeas1) (hY := hX\u2081) (hZ := hX\u2082)\n (Measurable.add hX\u2081' hX\u2082') (independenceCondition1 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n rw [\u2190 add_assoc, aux1] at aux2\n linarith [aux2]\n\n have ineq2 : d[X\u2080\u2082 # U | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := by\n have aux1 : H[S] + H[U] - H[X\u2082] - H[X\u2081' + X\u2082'] = H[S] - H[X\u2082] := by\n rw [hU X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep] ; ring\n have aux2 : d[X\u2080\u2082 # U | U + (X\u2081' + X\u2082')] - d[X\u2080\u2082 # X\u2082]\n \u2264 (H[U + (X\u2081' + X\u2082')] + H[U] - H[X\u2082] - H[X\u2081' + X\u2082']) / 2 := by\n rw [(show U = X\u2082 + X\u2081 from add_comm _ _)]\n apply condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas2) (hX\u2082) (hX\u2081)\n (Measurable.add hX\u2081' hX\u2082') (independenceCondition2 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n rw [\u2190 add_assoc, aux1] at aux2\n linarith [aux2]\n\n have V_add_eq : V + (X\u2081 + X\u2082') = S := by abel\n\n have ineq3 : d[X\u2080\u2081 # V | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 := by\n have aux2 : d[p.X\u2080\u2081 # V | V + (X\u2081 + X\u2082')] - d[p.X\u2080\u2081 # X\u2081']\n \u2264 (H[V + (X\u2081 + X\u2082')] + H[V] - H[X\u2081'] - H[X\u2081 + X\u2082']) / 2 :=\n condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas1) (hX\u2081') (hX\u2082) (Measurable.add hX\u2081 hX\u2082')\n (independenceCondition3 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n\n have aux1 : H[S] + H[V] - H[X\u2081'] - H[X\u2081 + X\u2082'] = H[S ; \u2119] - H[X\u2081 ; \u2119] := by\n rw [hV X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep, h\u2081.entropy_eq]; ring\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl p.hmeas1.aemeasurable) h\u2081,\n V_add_eq, aux1] at aux2\n linarith [aux2]\n\n have ineq4 : d[X\u2080\u2082 # V | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := by\n have aux2 : d[p.X\u2080\u2082 # V | V + (X\u2081 + X\u2082')] - d[p.X\u2080\u2082 # X\u2082]\n \u2264 (H[V + (X\u2081 + X\u2082')] + H[V] - H[X\u2082] - H[X\u2081 + X\u2082']) / 2 := by\n rw [(show V = X\u2082 + X\u2081' from add_comm _ _)]\n apply condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas2) (hX\u2082) (hX\u2081') (Measurable.add hX\u2081 hX\u2082')\n (independenceCondition4 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have aux1 : H[S] + H[V] - H[X\u2082] - H[X\u2081 + X\u2082'] = H[S ; \u2119] - H[X\u2082 ; \u2119] := by\n rw [hV X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep]; ring\n rw [V_add_eq, aux1] at aux2\n linarith [aux2]\n\n let W' := X\u2082 + X\u2082'\n have ineq5 : d[X\u2080\u2081 # W | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] + H[W ; \u2119] - H[X\u2081 ; \u2119] - H[W' ; \u2119])/2 := by\n have := condRuzsaDist_diff_ofsum_le \u2119 p.hmeas1 hX\u2081 hX\u2081' (Measurable.add hX\u2082 hX\u2082')\n (independenceCondition5 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have S_eq : X\u2081 + X\u2081' + (fun a \u21a6 X\u2082 a + X\u2082' a) = S := by\n rw [(show (fun a \u21a6 X\u2082 a + X\u2082' a) = X\u2082 + X\u2082' by rfl), \u2190 add_assoc, add_assoc X\u2081, add_comm X\u2081',\n \u2190 add_assoc]\n rwa [S_eq, add_comm X\u2081 X\u2081'] at this\n\n have ineq6 : d[X\u2080\u2082 # W' | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] + H[W' ; \u2119] - H[X\u2082 ; \u2119] - H[W ; \u2119])/2 := by\n have := condRuzsaDist_diff_ofsum_le \u2119 p.hmeas2 hX\u2082 hX\u2082' (Measurable.add hX\u2081' hX\u2081)\n (independenceCondition6 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have S_eq : X\u2082 + X\u2082' + (fun a \u21a6 X\u2081' a + X\u2081 a) = S := by\n rw [(show (fun a \u21a6 X\u2081' a + X\u2081 a) = X\u2081' + X\u2081 by rfl), add_comm, \u2190 add_assoc, add_comm X\u2081',\n add_assoc X\u2081, add_comm X\u2081', \u2190 add_assoc]\n rwa [S_eq] at this\n\n have dist_eq : d[X\u2080\u2082 # W' | S] = d[X\u2080\u2082 # W | S] := by\n have S_eq : S = (X\u2082 + X\u2082') + (X\u2081' + X\u2081) := by\n rw [add_comm X\u2081' X\u2081, add_assoc _ X\u2082', add_comm X\u2082', \u2190 add_assoc X\u2082, \u2190 add_assoc X\u2082,\n add_comm X\u2082]\n rw [S_eq]\n apply condRuzsaDist'_of_inj_map' p.hmeas2 (hX\u2082.add hX\u2082') (hX\u2081'.add hX\u2081)\n\n -- Put everything together to bound the sum of the `c` terms\n have ineq7 : c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264\n 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := by\n have step\u2081 : c[U|S # U|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc\n _ = (d[p.X\u2080\u2081 # U|S] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2082 # U|S] - d[p.X\u2080\u2082 # X\u2082]) := by ring\n _ \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 + (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := add_le_add ineq1 ineq2\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n have step\u2082 : c[V|S # V|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc c[V|S # V|S] =(d[p.X\u2080\u2081 # V|S] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2082 # V|S] - d[p.X\u2080\u2082 # X\u2082]) := by ring\n _ \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 + (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := add_le_add ineq3 ineq4\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n have step\u2083 : c[W|S # W|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc c[W|S # W|S] = (d[X\u2080\u2081 # W | S] - d[X\u2080\u2081 # X\u2081]) + (d[X\u2080\u2082 # W' | S] - d[X\u2080\u2082 # X\u2082]) :=\n by rw [dist_eq]\n _ \u2264 (H[S ; \u2119] + H[W ; \u2119] - H[X\u2081 ; \u2119] - H[W' ; \u2119])/2 + (H[S ; \u2119] + H[W' ; \u2119] - H[X\u2082 ; \u2119] - H[W ; \u2119])/2\n := add_le_add ineq5 ineq6\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) +\n (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) + (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) :=\n add_le_add (add_le_add step\u2081 step\u2082) step\u2083\n _ = 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := by ring\n\n have h_indep' : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'] := by\n apply ProbabilityTheory.iIndepFun.reindex (Equiv.swap (2 : Fin 4) 3)\n convert h_indep using 1\n ext x\n fin_cases x ; all_goals { aesop }\n\n have ineq8 : 3 * H[S ; \u2119] \u2264 3/2 * (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081 :=\n calc 3 * H[S ; \u2119] \u2264 3 * (H[X\u2081 ; \u2119] / 2 + H[X\u2082 ; \u2119] / 2 + (2+p.\u03b7)*k - I\u2081) := by\n apply (mul_le_mul_left (zero_lt_three' \u211d)).mpr\n (ent_ofsum_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep' h_min)\n _ = 3/2 * ( H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081 := by ring\n\n -- Final computation\n calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := ineq7\n _ = 3 * H[S ; \u2119] - (3/2 *(H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])) := by ring\n _ \u2264 (3/2 * ( H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081) - (3/2 *(H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])) :=\n sub_le_sub_right ineq8 _\n _ = (6 - 3 * p.\u03b7)*k + 3 * (2*p.\u03b7*k - I\u2081) := by ring", "proofType": "tactic", "proofLengthLines": 113, "proofLengthTokens": 6244}} +{"srcContext": "import Mathlib.Data.Set.Card\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Graph\nimport PFR.Mathlib.LinearAlgebra.Basis.VectorSpace\nimport PFR.Mathlib.SetTheory.Cardinal.Finite\n\n/-!\n# The homomorphism form of PFR\n\nHere we apply (improved) PFR to show that approximate homomorphisms f from a 2-group to a 2-group\nare close to actual homomorphisms. Here, approximate is in the sense that f(x+y)-f(x)-f(y) takes\nfew values.\n\n## Main results\n\n* `goursat`: A convenient description of the subspaces of $G \\times G'$ when $G, G'$ are elementary\n abelian 2-groups.\n* `homomorphism_pfr` : If $f : G \u2192 G'$ is a map between finite abelian elementary 2-groups such\n that $f(x+y)-f(x)-f(y)$ takes at most $K$ values, then there is a homomorphism $\\phi: G \\to G'$\n such that $f(x)-\\phi(x)$ takes at most $K^{12}$ values.\n-/\n\nopen Pointwise\n\nvariable {G G' : Type*} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G']\n [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2]\n\n/-- Let $H_0$ be a subgroup of $G$. Then every homomorphism $\\phi: H_0 \\to G'$ can be extended to a\nhomomorphism $\\tilde \\phi: G \\to G'$. -/\nlemma hahn_banach (H\u2080 : AddSubgroup G) (\u03c6 : H\u2080 \u2192+ G') : \u2203 (\u03c6' : G \u2192+ G'), \u2200 x : H\u2080, \u03c6 x = \u03c6' x := by\n let H\u2080 := AddSubgroup.toZModSubmodule 2 H\u2080\n let \u03c6 := (show H\u2080 \u2192+ G' from \u03c6).toZModLinearMap 2\n obtain \u27e8\u03c6', h\u03c6'\u27e9 := \u03c6.exists_extend\n use \u03c6'; intro x; show \u03c6 x = \u03c6'.comp H\u2080.subtype x; rw [h\u03c6']\n\n/-- Let $H$ be a subgroup of $G \\times G'$. Then there exists a subgroup $H_0$ of $G$, a\nsubgroup $H_1$ of $G'$, and a homomorphism $\\phi: G \\to G'$ such that\n$$ H := \\{ (x, \\phi(x) + y): x \\in H_0, y \\in H_1 \\}.$$\nIn particular, $|H| = |H_0| |H_1|$. -/\nlemma goursat (H : AddSubgroup (G \u00d7 G')): \u2203 (H\u2080 : AddSubgroup G) (H\u2081 : AddSubgroup G') (\u03c6 : G \u2192+ G'),\n (\u2200 x : G \u00d7 G', x \u2208 H \u2194 (x.1 \u2208 H\u2080 \u2227 x.2 - \u03c6 x.1 \u2208 H\u2081)) \u2227\n (Nat.card H) = (Nat.card H\u2080) * (Nat.card H\u2081) := by\n let H := AddSubgroup.toZModSubmodule 2 H\n obtain \u27e8S\u2081, S\u2082, f, \u03c6, hf, hf_inv\u27e9 := H.exists_equiv_fst_sndModFst\n use S\u2081.toAddSubgroup, S\u2082.toAddSubgroup, \u03c6\n constructor ; swap\n \u00b7 show Nat.card H = _\n exact Eq.trans (Nat.card_eq_of_bijective f f.bijective) (Nat.card_prod S\u2081 S\u2082)\n \u00b7 intro x\n \u00b7 constructor\n \u00b7 intro hx\n let x : H := { val := x, property := hx }\n \u00b7 constructor\n \u00b7 exact Set.mem_of_eq_of_mem (hf x).1.symm (f x).1.property\n \u00b7 exact Set.mem_of_eq_of_mem (hf x).2.symm (f x).2.property\n \u00b7 intro hx\n \u00b7 let x\u2081 : S\u2081 := { val := x.1, property := hx.1 }\n let x\u2082 : S\u2082 := { val := x.2 - \u03c6 x.1, property := hx.2 }\n exact Set.mem_of_eq_of_mem (by rw [hf_inv, sub_add_cancel]) (f.symm (x\u2081, x\u2082)).property\n\nopen Set Fintype\n\n", "theoremStatement": "/-- Let $f: G \\to G'$ be a function, and let $S$ denote the set\n$$ S := \\{ f(x+y)-f(x)-f(y): x,y \\in G \\}.$$\nThen there exists a homomorphism $\\phi: G \\to G'$ such that\n$$ |\\{f(x) - \\phi(x)\\}| \\leq |S|^{12}. $$ -/\ntheorem homomorphism_pfr (f : G \u2192 G') (S : Set G') (hS: \u2200 x y : G, f (x+y) - (f x) - (f y) \u2208 S) :\n \u2203 (\u03c6 : G \u2192+ G') (T : Set G'), Nat.card T \u2264 (Nat.card S) ^ 12 \u2227 \u2200 x : G, (f x) - (\u03c6 x) \u2208 T ", "theoremName": "homomorphism_pfr", "fileCreated": {"commit": 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"Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Compiler.CSimpAttr", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Tactic.Lemma", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Tactic.TypeStar", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Data.ULift", "Mathlib.Algebra.NeZero", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Tactic.Cases", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "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.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.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.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.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.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.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.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Finset.Option", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "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.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.Prod", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "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.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main", "PFR.ImprovedPFR", "PFR.ForMathlib.Graph", "PFR.Mathlib.LinearAlgebra.Basis.VectorSpace", "PFR.Mathlib.SetTheory.Cardinal.Finite"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n classical\n have : 0 < Nat.card G := Nat.card_pos\n let A := graph f\n have hA_le : (Nat.card \u21a5(A + A) : \u211d) \u2264 Nat.card S * Nat.card A := by\n let B := A - {0}\u00d7\u02e2S\n have hAB : A + A \u2286 B := by\n intro x hx\n obtain \u27e8a, ha, a', ha', haa'\u27e9 := Set.mem_add.mp hx\n simp only [mem_graph, A] at ha ha'\n rw [Set.mem_sub]\n refine \u27e8(x.1, f x.1), ?_, (0, f (a.1 + a'.1) - f a.1 - f a'.1), ?_\u27e9\n \u00b7 simp [A]\n \u00b7 simp only [singleton_prod, mem_image, Prod.mk.injEq, true_and,\n exists_eq_right, Prod.mk_sub_mk, sub_zero]\n exact \u27e8hS a.1 a'.1,\n by rw [\u2190 Prod.fst_add, ha, ha', sub_sub, \u2190 Prod.snd_add, haa', sub_sub_self]\u27e9\n have hB_card : Nat.card B \u2264 Nat.card S * Nat.card A :=\n card_sub_le.trans_eq $ by simp only [mul_comm, Set.card_singleton_prod]\n norm_cast\n exact (Nat.card_mono (toFinite B) hAB).trans hB_card\n have hA_nonempty : A.Nonempty := by use (0, f 0) ; exact \u27e80, rfl\u27e9\n obtain \u27e8H, c, hcS, -, -, hAcH\u27e9 := PFR_conjecture_improv_aux hA_nonempty hA_le\n have : 0 < Nat.card c := by\n have : c.Nonempty := by\n by_contra! H\n simp only [H, empty_add, subset_empty_iff] at hAcH\n simp [hAcH] at hA_nonempty\n exact this.card_pos c.toFinite\n obtain \u27e8H\u2080, H\u2081, \u03c6, hH\u2080\u2081, hH_card\u27e9 := goursat H\n have hG_card_le : Nat.card G \u2264 Nat.card c * Nat.card H\u2080 := by\n let c' := (Prod.fst) '' c\n have hc'_card : Nat.card c' \u2264 Nat.card c := Nat.card_image_le (toFinite c)\n have h_fstH : Prod.fst '' (H : Set (G \u00d7 G')) = (H\u2080 : Set G) := by\n ext x\n simp only [mem_image, SetLike.mem_coe, hH\u2080\u2081, Prod.exists,\n exists_and_right, exists_and_left, exists_eq_right, and_iff_left_iff_imp]\n exact fun _ \u21a6 \u27e8\u03c6 x, by simp only [sub_self, AddSubgroup.zero_mem]\u27e9\n have hG_cover : (univ : Set G) = c' + (H\u2080:Set G) := by\n apply (eq_univ_of_forall (fun g \u21a6 ?_)).symm\n have := image_subset Prod.fst hAcH\n rw [\u2190 AddHom.coe_fst, Set.image_add, AddHom.coe_fst, image_fst_graph] at this\n rw [\u2190 h_fstH]\n exact this (mem_univ g)\n apply_fun Nat.card at hG_cover\n rw [Nat.card_coe_set_eq, Set.ncard_univ] at hG_cover\n rw [hG_cover]\n calc\n (Nat.card (c'+ (H\u2080:Set G))) \u2264 Nat.card c' * Nat.card H\u2080 := card_add_le\n _ \u2264 Nat.card c * Nat.card H\u2080 := by gcongr\n have : (Nat.card H\u2081 : \u211d) \u2264 (Nat.card H / Nat.card A) * Nat.card c := by calc\n (Nat.card H\u2081 : \u211d) = (Nat.card H : \u211d) / Nat.card H\u2080 := by field_simp [hH_card, mul_comm]\n _ \u2264 (Nat.card H : \u211d) / (Nat.card G / Nat.card c) := by\n gcongr\n rw [div_le_iff' (by positivity)]\n exact_mod_cast hG_card_le\n _ = (Nat.card H / Nat.card G : \u211d) * Nat.card c := by field_simp\n _ = (Nat.card H / Nat.card A) * Nat.card c := by congr; exact (card_graph f).symm\n let T := (fun p \u21a6 p.2 - \u03c6 p.1) '' (c + {0} \u00d7\u02e2 (H\u2081: Set G'))\n have : A \u2286 \u22c3 (c \u2208 T), {(x, \u03c6 x + c) | x : G} := by\n have : (H : Set (G \u00d7 G')) \u2286 ({0} \u00d7\u02e2 (H\u2081:Set G')) + {(x, \u03c6 x) | x : G} := by\n rintro \u27e8g, g'\u27e9 hg\n simp only [SetLike.mem_coe, hH\u2080\u2081] at hg\n refine \u27e8(0, g' - \u03c6 g), ?_, (g, \u03c6 g), ?_\u27e9\n \u00b7 simp only [singleton_prod, mem_image, SetLike.mem_coe,\n Prod.mk.injEq, true_and, exists_eq_right, hg.2]\n \u00b7 simp only [mem_setOf_eq, Prod.mk.injEq, exists_eq_left, Prod.mk_add_mk, zero_add, true_and,\n sub_add_cancel]\n have hA_sub : A \u2286 c + (({0} \u00d7\u02e2 (H\u2081:Set G')) + {(x, \u03c6 x) | x : G}) :=\n calc\n A \u2286 c + (H : Set _) := hAcH\n _ \u2286 c + (({0} \u00d7\u02e2 (H\u2081:Set G')) + {(x, \u03c6 x) | x : G}) := add_subset_add_left this\n rw [\u2190 add_assoc] at hA_sub\n convert hA_sub\n rw [\u2190 Set.iUnion_add_left_image, \u2190 graph_def]\n simp_rw [graph_add, Set.biUnion_image]\n refine \u27e8\u03c6, T, ?_, ?_\u27e9\n \u00b7 have : (Nat.card T : \u211d) \u2264 (Nat.card S : \u211d) ^ (12 : \u211d) := by calc\n (Nat.card T:\u211d) \u2264 Nat.card (c + {(0:G)} \u00d7\u02e2 (H\u2081:Set G')) := by\n norm_cast; apply Nat.card_image_le (toFinite _)\n _ \u2264 Nat.card c * Nat.card H\u2081 := by\n norm_cast\n apply card_add_le.trans\n rw [Set.card_singleton_prod] ; rfl\n _ \u2264 Nat.card c * ((Nat.card H / Nat.card A) * Nat.card c) := by gcongr\n _ = (Nat.card c) ^ 2 * (Nat.card H / Nat.card A) := by ring\n _ \u2264 ((Nat.card S) ^ (6 : \u211d) * (Nat.card A) ^ (1 / 2 : \u211d) * (Nat.card H) ^ (-1 / 2 : \u211d)) ^ 2\n * (Nat.card H / Nat.card A) := by gcongr; exact hcS\n _ = (Nat.card S : \u211d) ^ (12 : \u211d) := by\n rw [\u2190 Real.rpow_two, div_eq_mul_inv, div_eq_mul_inv, div_eq_mul_inv]\n have : 0 < Nat.card S := by\n have : S.Nonempty := \u27e8f (0 + 0) - f 0 - f 0, hS 0 0\u27e9\n exact this.card_pos S.toFinite\n have : 0 < Nat.card A := hA_nonempty.card_pos A.toFinite\n have : 0 < Nat.card H := H.coe_nonempty.card_pos $ toFinite _\n rpow_ring\n norm_num\n exact_mod_cast this\n \u00b7 intro g\n specialize this (\u27e8g, rfl\u27e9 : (g, f g) \u2208 A)\n simp only [mem_iUnion, mem_setOf_eq, Prod.mk.injEq, exists_eq_left] at this\n obtain \u27e8t, \u27e8ht, h\u27e9\u27e9 := this\n rw [\u2190 h]\n convert ht\n abel", "proofType": "tactic", "proofLengthLines": 104, "proofLengthTokens": 4987}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] := by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]\n\n/-- Adding a constant to a random variable does not change the Rusza distance. -/\nlemma rdist_add_const [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y + fun _ \u21a6 c; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n have A : IdentDistrib (Y' + fun _ \u21a6 c) (Y + fun _ \u21a6 c) \u03bd \u03bc' := by\n change IdentDistrib (fun \u03c9 \u21a6 Y' \u03c9 + c) (fun \u03c9 \u21a6 Y \u03c9 + c) \u03bd \u03bc'\n apply hIdY.comp (measurable_add_const c)\n have B : IndepFun X' (Y' + fun _ \u21a6 c) \u03bd := by\n change IndepFun X' (fun \u03c9 \u21a6 Y' \u03c9 + c) \u03bd\n apply hind.comp measurable_id (measurable_add_const c)\n have C : X' - (Y' + fun _ \u21a6 c) = (X' - Y') + (fun _ \u21a6 -c) := by\n ext \u03c9; simp; abel\n rw [\u2190 hIdX.rdist_eq hIdY, \u2190 hIdX.rdist_eq A, hind.rdist_eq hX' hY',\n B.rdist_eq hX' (hY'.add_const _), entropy_add_const hY' c, C, entropy_add_const]\n exact hX'.sub hY'\n\n/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\nlemma rdist_add_const' [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] (c: G) (c': G)\n (hX : Measurable X) (hY : Measurable Y) :\n d[X + fun _ \u21a6 c; \u03bc # Y + fun _ \u21a6 c'; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_add_const _ hY, rdist_symm, rdist_add_const hY hX, rdist_symm]\n measurability\n\n/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (X : \u03a9 \u2192 G) (Y : \u03a9 \u2192 G) (Z : \u03a9 \u2192 G)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n have h1 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] + H[X - Y; \u03bc] \u2264 H[\u27e8X - Z, X - Y\u27e9; \u03bc] + H[\u27e8Y, X - Y\u27e9; \u03bc] :=\n entropy_triple_add_entropy_le \u03bc (hX.sub hZ) hY (hX.sub hY)\n have h2 : H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n calc H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n have : \u27e8X - Z, X - Y\u27e9 = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 \u27e8X - Z, Y - Z\u27e9 := by ext1; simp\n rw [this]\n apply entropy_comp_le \u03bc (by measurability)\n _ \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n have h : 0 \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] - H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n apply mutualInfo_nonneg (by measurability) (by measurability) \u03bc\n linarith\n have h3 : H[\u27e8Y, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X, Y\u27e9 ; \u03bc] := by\n have : \u27e8Y, X - Y\u27e9 = (fun p \u21a6 (p.2, p.1 - p.2)) \u2218 \u27e8X, Y\u27e9 := by ext1; simp\n rw [this]\n exact entropy_comp_le \u03bc (hX.prod_mk hY) _\n have h4 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] = H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] := by\n refine entropy_of_comp_eq_of_comp \u03bc ((hX.sub hZ).prod_mk (hY.prod_mk (hX.sub hY)))\n (hX.prod_mk (hY.prod_mk hZ))\n (fun p : G \u00d7 (G \u00d7 G) \u21a6 (p.2.2 + p.2.1, p.2.1, -p.1 + p.2.2 + p.2.1))\n (fun p : G \u00d7 G \u00d7 G \u21a6 (p.1 - p.2.2, p.2.1, p.1 - p.2.1)) ?_ ?_\n \u00b7 ext1; simp\n \u00b7 ext1; simp\n have h5 : H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [entropy_assoc hX hY hZ, entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h\n rw [h4, h5] at h1\n calc H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Y - Z; \u03bc] - H[Z; \u03bc] := by linarith\n _ = H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n congr 2\n rw [entropy_sub_comm hY hZ]\n\n/-- The **entropic Ruzsa triangle inequality** -/\nlemma rdist_triangle {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [h\u03bc : IsProbabilityMeasure \u03bc] [h\u03bc' : IsProbabilityMeasure \u03bc'] [h\u03bc'' : IsProbabilityMeasure \u03bc''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n d[X ; \u03bc # Z ; \u03bc''] \u2264 d[X ; \u03bc # Y ; \u03bc'] + d[Y ; \u03bc' # Z ; \u03bc''] := by\n obtain \u27e8A, mA, \u03bcA, X', Y', Z', h\u03bcA, hInd, hX', hY', hZ', HX, HY, HZ, _, _, _\u27e9 :=\n independent_copies3_nondep_finiteRange hX hY hZ \u03bc \u03bc' \u03bc''\n suffices d[X' ; \u03bcA # Z' ; \u03bcA] \u2264 d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] by\n rwa [HX.rdist_eq HY, HY.rdist_eq HZ, HX.rdist_eq HZ] at this\n have IndepLem : IndepFun (\u27e8X', Z'\u27e9) Y' \u03bcA :=\n iIndepFun.indepFun_prod_mk hInd (fun i => by fin_cases i ; all_goals { simpa }) 0 2 1\n (by norm_cast) (by norm_cast)\n calc d[X' ; \u03bcA # Z' ; \u03bcA] = H[X' - Z'; \u03bcA] - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) := by\n rw [ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using hInd.indepFun (show 0 \u2260 2 by norm_cast)) hX' hZ'] ; ring\n _ \u2264 (H[X' - Y' ; \u03bcA] + H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA]) - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) :=\n sub_le_sub_right (ent_of_diff_le _ _ _ hX' hZ' hY' IndepLem) _\n _ = (H[X' - Y' ; \u03bcA] - H[X'; \u03bcA] / 2 - H[Y' ; \u03bcA] / 2) +\n (H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA] / 2 - H[Z'; \u03bcA] / 2) := by ring\n _ = d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] := by\n rw [ProbabilityTheory.IndepFun.rdist_eq (by simpa using (iIndepFun.indepFun hInd\n (show 0 \u2260 1 by norm_cast))) hX' hY', ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using (iIndepFun.indepFun hInd (show 1 \u2260 2 by norm_cast))) hY' hZ']\n\nvariable [MeasurableSingletonClass S] [MeasurableSingletonClass T]\n\n/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) [IsFiniteMeasure \u03bc]\n (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc'\"]\" => condRuzsaDist X Z Y W \u03bc \u03bc'\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" # \" Y \" | \" W \"]\" => condRuzsaDist X Z Y W volume volume\n\nlemma condRuzsaDist_def (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W] := rfl\n\n/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\nlemma condRuzsaDist_symm {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hZ : Measurable Z)\n (hW : Measurable W) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange Z]\n [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[Y | W ; \u03bc' # X | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist_symm]\n\n@[simp] lemma condRuszaDist_zero_right (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n d[X | Z ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero,\n kernel.rdist_zero_right]\n\n@[simp] lemma condRuszaDist_zero_left (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; 0 # Y | W ; \u03bc'] = 0 := by\n simp [condRuzsaDist]\n\nlemma condRuzsaDist_nonneg {X : \u03a9 \u2192 G} (hX : Measurable X) [FiniteRange X]\n {Z : \u03a9 \u2192 S} (hZ : Measurable Z) [FiniteRange Z]\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) [FiniteRange Y]\n {W : \u03a9' \u2192 T} (hW : Measurable W) [FiniteRange W]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n 0 \u2264 d[X | Z ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def]\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n refine kernel.rdist_nonneg ?_ ?_\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hX hZ\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hY hW\n\n/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\nlemma rdist_eq_rdistm : d[X ; \u03bc # Y ; \u03bc'] = kernel.rdistm (\u03bc.map X) (\u03bc'.map Y) := rfl\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\nlemma condRuzsaDist_eq_sum {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z in FiniteRange.toFinset Z, \u2211 w in FiniteRange.toFinset W,\n (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((((FiniteRange.toFinset Z)\n \u00d7\u02e2 (FiniteRange.toFinset W)) : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Finset.sum_product,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\nlemma condRuzsaDist_eq_sum' {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [Fintype S] [Fintype T] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z, \u2211 w, (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Fintype.sum_prod_type,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist' (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc' \"]\" => condRuzsaDist' X Y W \u03bc \u03bc'\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" # \" Y \" | \" W \"]\" => condRuzsaDist' X Y W volume volume\n\n/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\nlemma condRuzsaDist'_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y | W ; \u03bc'] =\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W] :=\n rfl\n\n@[simp] lemma condRuzsaDist'_zero_right (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) :\n d[X ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist'_def, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero, kernel.rdist_zero_right]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\nlemma condRuzsaDist'_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (dirac ()) (\u03bc'.map W) ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 FiniteRange.toFinset W : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, smul_eq_mul, Finset.sum_product]\n simp only [Finset.univ_unique, PUnit.default_eq_unit, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, Pi.one_apply, one_mul,\n Finset.sum_singleton]\n simp_rw [Measure.map_apply hW (measurableSet_singleton _)]\n congr with w\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff]\n refine Or.inr ?_\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm, condDistrib_apply hY hW _ _ hw]\n congr\n\n/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\nlemma condRuzsaDist'_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_eq_sum hY hW \u03bc \u03bc']\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\nopen scoped ENNReal\n\nlemma condRuzsaDist'_prod_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [FiniteRange W] [FiniteRange W']:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n have : d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc'] = \u2211 w in ((FiniteRange.toFinset W') \u00d7\u02e2 FiniteRange.toFinset W), (\u03bc' ((fun a => (W' a, W a)) \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; \u03bc'[|(fun a => (W' a, W a)) \u207b\u00b9' {w}]] := by\n rw [condRuzsaDist'_eq_sum hY (hW'.prod_mk hW)]\n apply Finset.sum_subset\n . intro (t, t')\n simp\n intro \u03c9 h1 h2\n exact \u27e8\u27e8\u03c9, h1\u27e9, \u27e8\u03c9, h2\u27e9\u27e9\n intro (t, t') _ ht\n simp at ht\n have : (fun a => (W' a, W a)) \u207b\u00b9' {(t, t')} = \u2205 := by\n ext \u03c9\n simp\n exact ht \u03c9\n simp [this]\n rw [this, Finset.sum_product_right]\n congr 1 with w\n rw [condRuzsaDist'_eq_sum hY hW', Finset.mul_sum]\n congr 1 with w'\n rw [\u2190 mul_assoc]\n have A : (fun a \u21a6 (W' a, W a)) \u207b\u00b9' {(w', w)} = W' \u207b\u00b9' {w'} \u2229 W\u207b\u00b9' {w} := by ext; simp\n congr 1\n \u00b7 simp only [ProbabilityTheory.cond, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n smul_eq_mul, ENNReal.toReal_mul, A, restrict_apply (hW' (MeasurableSet.singleton w'))]\n rcases eq_bot_or_bot_lt (\u03bc' (W \u207b\u00b9' {w})) with hw|hw\n \u00b7 have : \u03bc' (W' \u207b\u00b9' {w'} \u2229 W \u207b\u00b9' {w}) = 0 :=\n le_antisymm (le_trans (measure_mono (Set.inter_subset_right _ _)) hw.le) bot_le\n simp [hw, this]\n \u00b7 rw [\u2190 mul_assoc, \u2190 ENNReal.toReal_mul, ENNReal.mul_inv_cancel, ENNReal.one_toReal, one_mul]\n exacts [hw.ne', by finiteness]\n \u00b7 congr 1\n rw [A, cond_cond_eq_cond_inter' _ (hW (.singleton w)) (hW' (.singleton w')), Set.inter_comm]\n finiteness\n\n/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\nlemma condRuzsaDist'_prod_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [Fintype T]:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_prod_eq_sum \u03bc \u03bc' hY hW' hW]\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\nlemma condRuzsaDist'_eq_integral (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = (\u03bc'.map W)[fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])]] := by\n rw [condRuzsaDist'_eq_sum hY hW]\n simp_rw [\u2190 smul_eq_mul]\n have : (\u03bc'.map W) (FiniteRange.toFinset W : Set T)\u1d9c = 0 := by\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n convert symm $ integral_eq_sum' (\u03bc'.map W) this (fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])])\n rw [Measure.map_apply hW (MeasurableSet.singleton _)]\n\n/-- Conditioning by a constant does not affect Ruzsa distance. -/\nlemma condRuzsaDist_of_const {X : \u03a9 \u2192 G} (hX : Measurable X) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (c : S)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange W] :\n d[X|(fun _ \u21a6 c) ; \u03bc # Y | W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def, condRuzsaDist'_def, Measure.map_const,measure_univ,one_smul, kernel.rdist,\n kernel.rdist, integral_prod, integral_dirac, integral_prod,integral_dirac]\n dsimp; congr; ext x; congr\n rw [condDistrib_apply hX measurable_const]\n \u00b7 simp\n \u00b7 simp\n \u00b7 exact integrable_of_finiteSupport _\n \u00b7 exact integrable_of_finiteSupport _\n\n/-- If $(X,Z)$ and $(Y,W)$ are independent, then\n$$ d[X | Z ; Y | W] = H[X'- Y'|Z', W'] - H[X'|Z']/2 - H[Y'|W']/2$$\n-/\nlemma condRuzsaDist_of_indep\n {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun (\u27e8X, Z\u27e9) (\u27e8Y, W\u27e9) \u03bc) [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc] = H[X - Y | \u27e8Z, W\u27e9 ; \u03bc] - H[X | Z ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ (hZ.prod_mk hW),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hW]\n swap; \u00b7 exact hX.sub hY\n congr 2\n have hZW : IndepFun Z W \u03bc := by\n have h' := IndepFun.comp h measurable_snd measurable_snd\n exact h'\n have hZW_map : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map Z).prod (\u03bc.map W) :=\n (indepFun_iff_map_prod_eq_prod_map_map hZ.aemeasurable hW.aemeasurable).mp hZW\n rw [\u2190 hZW_map]\n refine kernel.entropy_congr ?_\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (hZ.prod_mk hW) _ _).symm\n refine (this.symm.trans ?_).symm\n suffices kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft S (condDistrib Y W \u03bc)\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc by\n filter_upwards [this] with x hx\n rw [kernel.map_apply, kernel.map_apply, hx]\n . exact (condDistrib_eq_prod_of_indepFun hX hZ hY hW \u03bc h).symm\n\n", "theoremStatement": "/-- Formula for conditional Ruzsa distance for independent sets of variables. -/\nlemma condRuzsaDist'_of_indep {X : \u03a9 \u2192 G} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun X (\u27e8Y, W\u27e9) \u03bc) [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc] = H[X - Y | W ; \u03bc] - H[X ; \u03bc]/2 - H[Y | W ; \u03bc]/2 ", "theoremName": "condRuzsaDist'_of_indep", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc621d65f3d5b43306793549b7af0b5c922d3", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 713, "tokenPositionInFile": 36408, "theoremPositionInFile": 48}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 20, "numPremises": 184, "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", 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"Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist'_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ hW,\n condEntropy_eq_kernel_entropy hY hW, entropy_eq_kernel_entropy]\n rotate_left\n \u00b7 exact hX.sub hY\n congr 2\n let Z : \u03a9 \u2192 Unit := fun _ \u21a6 ()\n rw [\u2190 condDistrib_unit_right hX \u03bc]\n have h' : IndepFun (\u27e8X,Z\u27e9) (\u27e8Y, W\u27e9) \u03bc := by\n rw [indepFun_iff_measure_inter_preimage_eq_mul]\n intro s t hs ht\n have : \u27e8X, Z\u27e9 \u207b\u00b9' s = X \u207b\u00b9' ((fun c \u21a6 (c, ())) \u207b\u00b9' s) := by ext1 y; simp\n rw [this]\n rw [indepFun_iff_measure_inter_preimage_eq_mul] at h\n exact h _ _ (measurable_prod_mk_right hs) ht\n have h_indep := condDistrib_eq_prod_of_indepFun hX measurable_const hY hW _ h'\n have h_meas_eq : \u03bc.map (\u27e8Z, W\u27e9) = (Measure.dirac ()).prod (\u03bc.map W) := by\n ext s hs\n rw [Measure.map_apply (measurable_const.prod_mk hW) hs, Measure.prod_apply hs, lintegral_dirac,\n Measure.map_apply hW (measurable_prod_mk_left hs)]\n congr\n rw [\u2190 h_meas_eq]\n have : kernel.map (kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft Unit (condDistrib Y W \u03bc)) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc)\n (fun x \u21a6 x.1 - x.2) measurable_sub := by\n filter_upwards [h_indep] with y hy\n conv_rhs => rw [kernel.map_apply, hy]\n rw [kernel.entropy_congr this]\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (measurable_const.prod_mk hW) _ _).symm\n rw [kernel.entropy_congr this]\n have h_meas : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map W).map (Prod.mk ()) := by\n ext s hs\n rw [Measure.map_apply measurable_prod_mk_left hs, h_meas_eq, Measure.prod_apply hs,\n lintegral_dirac]\n have h_ker : condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.prodMkLeft Unit (condDistrib (X - Y) W \u03bc) := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply (measurable_const.prod_mk hW) (measurableSet_singleton _)] at hx\n ext s hs\n have h_preimage_eq : (fun a \u21a6 (PUnit.unit, W a)) \u207b\u00b9' {x} = W \u207b\u00b9' {x.2} := by\n conv_lhs => rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod]\n ext1 y\n simp\n rw [kernel.prodMkLeft_apply, condDistrib_apply' _ (measurable_const.prod_mk hW) _ _ hx hs,\n condDistrib_apply' _ hW _ _ _ hs]\n rotate_left\n \u00b7 exact hX.sub hY\n \u00b7 convert hx\n exact h_preimage_eq.symm\n \u00b7 exact hX.sub hY\n congr\n rw [kernel.entropy_congr h_ker, h_meas, kernel.entropy_prodMkLeft_unit]", "proofType": "tactic", "proofLengthLines": 56, "proofLengthTokens": 2673}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.Group\nimport PFR.Mathlib.MeasureTheory.Measure.MeasureSpace\n\n/-!\n# Ruzsa distance between kernels\n\n## Main definitions\n\n*\n\n## Notations\n\n* `dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] = `\n\n-/\n\n\nopen Real MeasureTheory\n\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\n\nnamespace ProbabilityTheory.kernel\n\nvariable {T T' T'' G : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T']\n [Countable T''] [Nonempty T''] [MeasurableSpace T''] [MeasurableSingletonClass T'']\n [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup G] [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G]\n {\u03bc : Measure T} {\u03bd : Measure T'}\n\n/-- The Rusza distance between two measures, defined as `H[X - Y] - H[X]/2 - H[Y]/2` where `X`\nand `Y` are independent variables distributed according to the two measures. -/\nnoncomputable\ndef rdistm (\u03bc : Measure G) (\u03bd : Measure G) : \u211d :=\n Hm[(\u03bc.prod \u03bd).map (fun x \u21a6 x.1 - x.2)] - Hm[\u03bc]/2 - Hm[\u03bd]/2\n\n/-- The Rusza distance between two kernels taking values in the same space, defined as the average\nRusza distance between the image measures. -/\nnoncomputable\ndef rdist (\u03ba : kernel T G) (\u03b7 : kernel T' G) (\u03bc : Measure T) (\u03bd : Measure T') : \u211d :=\n (\u03bc.prod \u03bd)[fun p \u21a6 rdistm (\u03ba p.1) (\u03b7 p.2)]\n\n/-- The Rusza distance between two kernels taking values in the same space, defined as the average\nRusza distance between the image measures. -/\nnotation3:max \"dk[\" \u03ba \" ; \" \u03bc \" # \" \u03b7 \" ; \" \u03bc' \"]\" => rdist \u03ba \u03b7 \u03bc \u03bc'\n\nlemma rdist_eq {\u03ba : kernel T G} {\u03b7 : kernel T' G} {\u03bc : Measure T} {\u03bd : Measure T'}\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd] :\n dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] = (\u03bc.prod \u03bd)[fun p \u21a6 Hm[((\u03ba p.1).prod (\u03b7 p.2)).map (fun x \u21a6 x.1 - x.2)]]\n - Hk[\u03ba, \u03bc]/2 - Hk[\u03b7, \u03bd]/2 := by\n rw [rdist]\n simp_rw [rdistm]\n rw [integral_sub, integral_sub]\n \u00b7 simp_rw [div_eq_mul_inv, integral_mul_right, integral_prod _ (integrable_of_finiteSupport _)]\n simp [entropy]\n all_goals { exact integrable_of_finiteSupport _ }\n\nlemma rdist_eq' {\u03ba : kernel T G} {\u03b7 : kernel T' G} [IsFiniteKernel \u03ba] [IsFiniteKernel \u03b7]\n {\u03bc : Measure T} {\u03bd : Measure T'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd] :\n dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] =\n Hk[map ((prodMkRight T' \u03ba) \u00d7\u2096 (prodMkLeft T \u03b7)) (fun x \u21a6 x.1 - x.2) measurable_sub, \u03bc.prod \u03bd]\n - Hk[\u03ba, \u03bc]/2 - Hk[\u03b7, \u03bd]/2 := by\n rw [rdist_eq]\n congr with p\n simp only\n rw [map_apply, prod_apply, prodMkLeft_apply, prodMkRight_apply]\n\nlemma rdist_symm {\u03ba : kernel T G} {\u03b7 : kernel T' G} [IsFiniteKernel \u03ba] [IsFiniteKernel \u03b7]\n {\u03bc : Measure T} {\u03bd : Measure T'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd] :\n dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] = dk[\u03b7 ; \u03bd # \u03ba ; \u03bc] := by\n rw [rdist_eq', rdist_eq', sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_comap_swap, comap_map_comm, entropy_sub_comm, Measure.comap_swap, Measure.prod_swap,\n comap_prod_swap, map_map]\n congr\n\n@[simp] lemma rdist_zero_right (\u03ba : kernel T G) (\u03b7 : kernel T' G) (\u03bc : Measure T) :\n dk[\u03ba ; \u03bc # \u03b7 ; 0] = 0 := by\n simp [rdist]\n\n@[simp] lemma rdist_zero_left (\u03ba : kernel T G) (\u03b7 : kernel T' G) (\u03bd' : Measure T') :\n dk[\u03ba ; 0 # \u03b7 ; \u03bd'] = 0 := by\n simp [rdist]\n\n@[simp] lemma rdist_zero_kernel_right {\u03ba : kernel T G} [IsFiniteKernel \u03ba]\n {\u03bc : Measure T} {\u03bd : Measure T'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd] :\n dk[\u03ba ; \u03bc # 0 ; \u03bd] = - Hk[\u03ba, \u03bc] / 2 := by\n rw [rdist_eq']\n simp only [prodMkLeft_zero, entropy_zero_kernel, zero_div, sub_zero]\n rw [sub_eq_iff_eq_add]\n ring_nf\n have : map (prodMkRight T' \u03ba \u00d7\u2096 0) (fun x \u21a6 x.1 - x.2) measurable_sub\n = 0 := by\n ext1 x\n rw [map_apply, prod_apply]\n simp\n rw [this, entropy_zero_kernel]\n\n@[simp] lemma rdist_zero_kernel_left {\u03b7 : kernel T' G} [IsFiniteKernel \u03b7]\n {\u03bc : Measure T} {\u03bd : Measure T'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd] :\n dk[0 ; \u03bc # \u03b7 ; \u03bd] = - Hk[\u03b7, \u03bd] / 2 := by\n rw [rdist_symm, rdist_zero_kernel_right]\n\n@[simp] lemma rdist_dirac_zero_right {\u03ba : kernel T G} [IsFiniteKernel \u03ba]\n {\u03bc : Measure T} {\u03bd : Measure T'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd] :\n dk[\u03ba ; \u03bc # const T' (Measure.dirac 0) ; \u03bd] = Hk[\u03ba, \u03bc] / 2 := by\n rw [rdist_eq']\n simp only [entropy_const, measure_univ, ENNReal.one_toReal, measureEntropy_dirac, mul_zero,\n zero_div, sub_zero]\n rw [sub_eq_iff_eq_add]\n ring_nf\n have : map (prodMkRight T' \u03ba \u00d7\u2096 prodMkLeft T (const T' (Measure.dirac 0)))\n (fun x \u21a6 x.1 - x.2) measurable_sub\n = prodMkRight T' \u03ba := by\n ext x s hs\n rw [prodMkRight_apply, map_apply, prod_apply, prodMkLeft_apply, const_apply,\n prodMkRight_apply, Measure.map_apply measurable_sub hs,\n Measure.prod_apply (measurable_sub hs)]\n simp only [Set.mem_preimage, sub_zero, Measure.dirac_apply]\n have : \u2200 x : G, Prod.mk x \u207b\u00b9' ((fun p : G \u00d7 G \u21a6 p.1 - p.2) \u207b\u00b9' s) = {y | x - y \u2208 s} := by\n intro x\n ext y\n simp\n simp_rw [this]\n have : \u2200 x : G, Set.indicator {y | x - y \u2208 s} (1 : G \u2192 \u211d\u22650\u221e) 0 = s.indicator (fun _ \u21a6 1) x := by\n intro x\n by_cases hx : x \u2208 s <;> simp [hx]\n simp_rw [this]\n rw [lintegral_indicator_const hs, one_mul]\n rw [this, entropy_prodMkRight']\n\n@[simp] lemma rdist_dirac_zero_left {\u03b7 : kernel T' G} [IsFiniteKernel \u03b7]\n {\u03bc : Measure T} {\u03bd : Measure T'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd] :\n dk[const T (Measure.dirac 0) ; \u03bc # \u03b7 ; \u03bd] = Hk[\u03b7, \u03bd] / 2 := by\n rw [rdist_symm, rdist_dirac_zero_right]\n\nlemma ruzsa_triangle_aux (\u03ba : kernel T (G \u00d7 G)) (\u03b7 : kernel T G)\n [IsMarkovKernel \u03ba] [IsMarkovKernel \u03b7] :\n map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.2 - p.1.2) (measurable_of_countable _)\n = map (\u03b7 \u00d7\u2096 snd \u03ba) (fun p \u21a6 p.1 - p.2) (measurable_of_countable _) := by\n have : (fun p : G \u00d7 G \u21a6 p.1 - p.2) = (fun p \u21a6 p.2 - p.1) \u2218 Prod.swap := by ext1 p; simp\n rw [this, \u2190 map_map _ measurable_swap (measurable_of_countable _), map_prod_swap]\n ext x s hs\n rw [map_apply' _ _ _ hs, map_apply' _ _ _ hs, prod_apply' _ _ _ (measurable_of_countable _ hs),\n prod_apply' _ _ _ (measurable_of_countable _ hs), lintegral_snd _ _ (measurable_of_countable _)]\n congr\n\nlemma abs_sub_entropy_le_rdist {\u03ba : kernel T G} {\u03b7 : kernel T' G}\n [IsMarkovKernel \u03ba] [IsMarkovKernel \u03b7]\n {\u03bc : Measure T} {\u03bd : Measure T'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7 : AEFiniteKernelSupport \u03b7 \u03bd) :\n |Hk[\u03ba, \u03bc] - Hk[\u03b7, \u03bd]| \u2264 2 * dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] := by\n have h := max_entropy_le_entropy_sub_prod (prodMkRight T' \u03ba) (prodMkLeft T \u03b7) (\u03bc.prod \u03bd)\n (h\u03ba.prodMkRight \u03bd) (h\u03b7.prodMkLeft \u03bc)\n rw [entropy_prodMkRight', entropy_prodMkLeft] at h\n rw [rdist_eq', abs_le]\n constructor\n \u00b7 linarith [le_max_right (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bd])]\n \u00b7 linarith [le_max_left (Hk[\u03ba, \u03bc]) (Hk[\u03b7, \u03bd])]\n\nlemma rdist_nonneg {\u03ba : kernel T G} {\u03b7 : kernel T' G}\n [IsMarkovKernel \u03ba] [IsMarkovKernel \u03b7]\n {\u03bc : Measure T} {\u03bd : Measure T'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7 : AEFiniteKernelSupport \u03b7 \u03bd) :\n 0 \u2264 dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] := by\n suffices 0 \u2264 2 * dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] by linarith\n calc 0 \u2264 |Hk[\u03ba, \u03bc] - Hk[\u03b7, \u03bd]| := abs_nonneg _\n _ \u2264 2 * dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] := abs_sub_entropy_le_rdist h\u03ba h\u03b7\n\n-- Kernel equivalent of `H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc]`\n-- `\u03ba` is `\u27e8X,Y\u27e9`, `\u03b7` is `Z`. Independence is expressed through the product `\u00d7\u2096`.\n", "theoremStatement": "/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (\u03ba : kernel T (G \u00d7 G)) (\u03b7 : kernel T G) [IsMarkovKernel \u03ba] [IsMarkovKernel \u03b7]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : FiniteKernelSupport \u03ba) (h\u03b7 : FiniteKernelSupport \u03b7) :\n Hk[map \u03ba (fun p : G \u00d7 G \u21a6 p.1 - p.2) measurable_sub, \u03bc]\n \u2264 Hk[map ((fst \u03ba) \u00d7\u2096 \u03b7) (fun p : G \u00d7 G \u21a6 p.1 - p.2) measurable_sub, \u03bc]\n + Hk[map (\u03b7 \u00d7\u2096 (snd \u03ba)) (fun p : G \u00d7 G \u21a6 p.1 - p.2) measurable_sub, \u03bc]\n - Hk[\u03b7, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.ent_of_diff_le", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "61fbca1a5afcf0aede80c8e0f215fe0488b6c455", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 183, "tokenPositionInFile": 7764, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 18, "numPremises": 243, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", 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"Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "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", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "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", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.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.Lemma", "Mathlib.Tactic.TypeStar", "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.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.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.Init.Data.Int.Basic", "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.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "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.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.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.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "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.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.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h\u03ba\u03b7 := kernel.finiteKernelSupport_of_prod h\u03ba h\u03b7\n have h1 : Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 (p.1.1 - p.2, (p.1.2, p.1.1 - p.1.2)))\n (measurable_of_countable _), \u03bc] + Hk[map \u03ba (fun p \u21a6 p.1 - p.2) measurable_sub, \u03bc]\n \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 (p.1.1 - p.2, p.1.1 - p.1.2)) (measurable_of_countable _), \u03bc]\n + Hk[map \u03ba (fun p \u21a6 (p.2, p.1 - p.2)) (measurable_of_countable _), \u03bc] := by\n have h := entropy_triple_add_entropy_le\n (map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 (p.1.1 - p.2, (p.1.2, p.1.1 - p.1.2))) (measurable_of_countable _)) \u03bc\n simp only [snd_map_prod _ (measurable_of_countable _) (measurable_of_countable _)] at h\n rw [deleteMiddle_map_prod _ (measurable_of_countable _) (measurable_of_countable _)\n (measurable_of_countable _)] at h\n have : map (\u03ba \u00d7\u2096 \u03b7) (fun x \u21a6 x.1.1 - x.1.2) (measurable_of_countable _)\n = map \u03ba (fun p \u21a6 p.1 - p.2) measurable_sub := by\n have : (fun x : (G \u00d7 G) \u00d7 G \u21a6 x.1.1 - x.1.2)\n = (fun x \u21a6 x.1 - x.2) \u2218 Prod.fst := by ext1 y; simp\n rw [this, \u2190 map_map, \u2190 kernel.fst, fst_prod]\n rw [this] at h\n refine (h ?_).trans_eq ?_\n . apply FiniteKernelSupport.aefiniteKernelSupport\n apply kernel.finiteKernelSupport_of_map\n exact kernel.finiteKernelSupport_of_prod h\u03ba h\u03b7\n congr 2\n have : (fun x : (G \u00d7 G) \u00d7 G \u21a6 (x.1.2, x.1.1 - x.1.2))\n = (fun x \u21a6 (x.2, x.1 - x.2)) \u2218 Prod.fst := by ext1 y; simp\n rw [this, \u2190 map_map, \u2190 kernel.fst, fst_prod]\n have h2 : Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 (p.1.1 - p.2, p.1.1 - p.1.2)) (measurable_of_countable _), \u03bc]\n \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1.1 - p.2) (measurable_of_countable _), \u03bc]\n + Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1.2 - p.2) (measurable_of_countable _), \u03bc] := by\n calc Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 (p.1.1 - p.2, p.1.1 - p.1.2)) (measurable_of_countable _), \u03bc]\n \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 (p.1.1 - p.2, p.1.2 - p.2)) (measurable_of_countable _), \u03bc] := by\n have : (fun p : (G \u00d7 G) \u00d7 G \u21a6 (p.1.1 - p.2, p.1.1 - p.1.2))\n = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 (fun p \u21a6 (p.1.1 - p.2, p.1.2 - p.2)) := by ext1; simp\n rw [this, \u2190 map_map]\n apply entropy_map_le _ _\n apply FiniteKernelSupport.aefiniteKernelSupport\n apply kernel.finiteKernelSupport_of_map h\u03ba\u03b7\n _ \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1.1 - p.2) (measurable_of_countable _), \u03bc]\n + Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1.2 - p.2) (measurable_of_countable _), \u03bc] := by\n have h : 0 \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1.1 - p.2) (measurable_of_countable _), \u03bc]\n + Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1.2 - p.2) (measurable_of_countable _), \u03bc]\n - Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 (p.1.1 - p.2, p.1.2 - p.2))\n (measurable_of_countable _), \u03bc] := by\n have h' := mutualInfo_nonneg (\u03ba := map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 (p.1.1 - p.2, p.1.2 - p.2))\n (measurable_of_countable _)) (\u03bc := \u03bc) ?_\n rwa [mutualInfo, fst_map_prod _ (measurable_of_countable _) (measurable_of_countable _),\n snd_map_prod _ (measurable_of_countable _) (measurable_of_countable _)] at h'\n apply FiniteKernelSupport.aefiniteKernelSupport\n apply kernel.finiteKernelSupport_of_map h\u03ba\u03b7\n linarith\n have h3 : Hk[map \u03ba (fun p : G \u00d7 G \u21a6 (p.2, p.1 - p.2)) (measurable_of_countable _), \u03bc]\n \u2264 Hk[\u03ba, \u03bc] := by\n exact entropy_map_le _ (h\u03ba.aefiniteKernelSupport _)\n have h4 : Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 (p.1.1 - p.2, (p.1.2, p.1.1 - p.1.2)))\n (measurable_of_countable _), \u03bc] = Hk[\u03ba \u00d7\u2096 \u03b7, \u03bc] := by\n refine entropy_of_map_eq_of_map\n (fun p : G \u00d7 G \u00d7 G \u21a6 ((p.2.2 + p.2.1, p.2.1), -p.1 + p.2.2 + p.2.1))\n (fun p : (G \u00d7 G) \u00d7 G \u21a6 (p.1.1 - p.2, (p.1.2, p.1.1 - p.1.2))) ?_ ?_ ?_\n (h\u03ba\u03b7.aefiniteKernelSupport _)\n \u00b7 rw [map_map]\n suffices ((fun p : G \u00d7 G \u00d7 G \u21a6 ((p.2.2 + p.2.1, p.2.1), -p.1 + p.2.2 + p.2.1))\n \u2218 fun p \u21a6 (p.1.1 - p.2, p.1.2, p.1.1 - p.1.2)) = id by\n simp_rw [this, map_id]\n ext1 p\n simp\n \u00b7 rfl\n apply FiniteKernelSupport.aefiniteKernelSupport\n apply kernel.finiteKernelSupport_of_map h\u03ba\u03b7\n have h5 : Hk[\u03ba \u00d7\u2096 \u03b7, \u03bc] = Hk[\u03ba, \u03bc] + Hk[\u03b7, \u03bc] := by\n rw [entropy_prod (h\u03ba.aefiniteKernelSupport _) (h\u03b7.aefiniteKernelSupport _)]\n rw [h4, h5] at h1\n calc Hk[map \u03ba (fun p : G \u00d7 G \u21a6 p.1 - p.2) measurable_sub, \u03bc]\n \u2264 Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1.1 - p.2) (measurable_of_countable _), \u03bc]\n + Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1.2 - p.2) (measurable_of_countable _), \u03bc] - Hk[\u03b7, \u03bc] := by\n linarith\n _ = Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.1.1 - p.2) (measurable_of_countable _), \u03bc]\n + Hk[map (\u03ba \u00d7\u2096 \u03b7) (fun p \u21a6 p.2 - p.1.2) (measurable_of_countable _), \u03bc] - Hk[\u03b7, \u03bc] := by\n congr 2\n rw [\u2190 entropy_neg, map_map]\n congr with p\n simp\n _ = Hk[map ((fst \u03ba) \u00d7\u2096 \u03b7) (fun p : G \u00d7 G \u21a6 p.1 - p.2) measurable_sub, \u03bc]\n + Hk[map (\u03b7 \u00d7\u2096 (snd \u03ba)) (fun p : G \u00d7 G \u21a6 p.1 - p.2) measurable_sub, \u03bc]\n - Hk[\u03b7, \u03bc] := by\n congr 3\n \u00b7 ext x s hs\n rw [map_apply' _ _ _ hs, map_apply' _ _ _ hs, prod_apply', prod_apply', lintegral_fst]\n \u00b7 congr with x\n \u00b7 exact measurable_of_countable _\n \u00b7 exact measurable_sub hs\n \u00b7 exact measurable_of_countable _ hs\n \u00b7 exact ruzsa_triangle_aux \u03ba \u03b7", "proofType": "tactic", "proofLengthLines": 90, "proofLengthTokens": 5227}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\n", "theoremStatement": "lemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 ", "theoremName": "gen_ineq_aux1", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "7b9dc64d2818953df55b83e56a95ceac2d68457e", "date": "2023-12-13"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 34, "tokenPositionInFile": 802, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 17, "numPremises": 315, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", 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"Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith", "proofType": "tactic", "proofLengthLines": 35, "proofLengthTokens": 2076}} +{"srcContext": "import PFR.TauFunctional\nimport PFR.HundredPercent\nimport PFR.Endgame\n\n/-!\n# Entropic version of polynomial Freiman-Ruzsa conjecture\n\nHere we prove the entropic version of the polynomial Freiman-Ruzsa conjecture.\n\n## Main results\n\n* `entropic_PFR_conjecture`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\n subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 11 d[X^0_1;X^0_2]$.\n\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n\nvariable {\u03a9 \u03a9' : Type*} [m\u03a9 : MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n [MeasurableSingletonClass G]\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G) {X\u2081 : \u03a9 \u2192 G} {X\u2082 : \u03a9 \u2192 G} (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082)\n\n/-- If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that $\\tau[X'_1;X'_2] < \\tau[X_1;X_2]$.\nPhrased in the contrapositive form for convenience of proof. -/\ntheorem tau_strictly_decreases (h_min : tau_minimizes p X\u2081 X\u2082) (hp\u03b7: p.\u03b7 = 1/9): d[X\u2081 # X\u2082] = 0 := by\n let \u27e8A, mA, \u03bc, Y\u2081, Y\u2082, Y\u2081', Y\u2082', h\u03bc, h_indep, hY\u2081, hY\u2082, hY\u2081', hY\u2082', h_id1, h_id2, h_id1', h_id2'\u27e9\n := independent_copies4_nondep hX\u2081 hX\u2082 hX\u2081 hX\u2082 \u2119 \u2119 \u2119 \u2119\n rw [\u2190 h_id1.rdist_eq h_id2]\n letI : MeasureSpace A := \u27e8\u03bc\u27e9\n have : IsProbabilityMeasure (\u2119 : Measure A) := h\u03bc\n rw [\u2190 h_id1.tau_minimizes p h_id2] at h_min\n apply tau_strictly_decreases_aux p Y\u2081 Y\u2082 Y\u2081' Y\u2082' hY\u2081 hY\u2082 hY\u2081' hY\u2082' (h_id1.trans h_id1'.symm)\n (h_id2.trans h_id2'.symm) h_indep h_min hp\u03b7\n\n\n", "theoremStatement": "/-- `entropic_PFR_conjecture`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\n subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 11 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture (hp\u03b7 : p.\u03b7 = 1/9):\n \u2203 H : AddSubgroup G, \u2203 \u03a9 : Type uG, \u2203 m\u03a9 : MeasureSpace \u03a9, \u2203 U : \u03a9 \u2192 G,\n IsProbabilityMeasure (\u2119 : Measure \u03a9) 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"Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8\u03a9', m\u03a9', X\u2081, X\u2082, hX\u2081, hX\u2082, _, htau_min\u27e9 := tau_minimizer_exists p\n have hdist : d[X\u2081 # X\u2082] = 0 := tau_strictly_decreases p hX\u2081 hX\u2082 htau_min hp\u03b7\n obtain \u27e8H, U, hU, hH_unif, hdistX\u2081, hdistX\u2082\u27e9 := exists_isUniform_of_rdist_eq_zero hX\u2081 hX\u2082 hdist\n refine \u27e8H, \u03a9', inferInstance, U, inferInstance, hU, hH_unif , ?_\u27e9\n have h : \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[p.X\u2080\u2082 # p.X\u2080\u2081 | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1\n rw [tau, tau, hp\u03b7] at h\n norm_num at h\n have : d[p.X\u2080\u2081 # p.X\u2080\u2082] = d[p.X\u2080\u2082 # p.X\u2080\u2081] := rdist_symm\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # X\u2081] + d[X\u2081 # U] := rdist_triangle p.hmeas1 hX\u2081 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # X\u2082] + d[X\u2082 # U] := rdist_triangle p.hmeas2 hX\u2082 hU\n linarith", "proofType": "tactic", "proofLengthLines": 11, "proofLengthTokens": 698}} +{"srcContext": "import PFR.FirstEstimate\nimport PFR.Mathlib.Probability.Independence.FourVariables\n\n/-!\n# Second estimate\n\nThe second estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ be independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $d[X_1;X_2] = k$\n* $I_1 := I_1 [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n* $I_2 := I[X_1+X_2 : X_1 + \\tilde X_1 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `second_estimate` : $$ I_2 \\leq 2 \\eta k + \\frac{2 \\eta (2 \\eta k - I_1)}{1 - \\eta}.$$\n-/\n\nopen MeasureTheory ProbabilityTheory ElementaryAddCommGroup\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2081', X\u2082'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n", "theoremStatement": "/-- $$ d[X_1+\\tilde X_1; X_2+\\tilde X_2] \\geq k - \\frac{\\eta}{2} ( d[X_1; X_1] + d[X_2;X_2] ).$$\n-/\nlemma rdist_of_sums_ge' : d[X\u2081 + X\u2081' # X\u2082 + X\u2082'] \u2265 k - p.\u03b7 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082]) / 2 ", "theoremName": "rdist_of_sums_ge'", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "8f6c9372ebe914cf676c0c44a6770933ad44ffdf", "date": "2023-11-22"}, "file": "PFR/PFR/SecondEstimate.lean", "module": "PFR.SecondEstimate", "jsonFile": "PFR.SecondEstimate.jsonl", 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"Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n refine' LE.le.ge (LE.le.trans _ (distance_ge_of_min p h_min (hX\u2081.add hX\u2081') (hX\u2082.add hX\u2082')))\n rw [sub_sub, sub_le_sub_iff_left k, \u2190 mul_add,mul_div_assoc]\n refine' mul_le_mul_of_nonneg_left _ (by linarith [p.h\u03b7])\n have h\u2081' := condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2081' (h_indep.indepFun (show 0 \u2260 2 by decide))\n have h\u2082' := condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2082' (h_indep.indepFun (show 1 \u2260 3 by decide))\n rw [h\u2081.entropy_eq, add_sub_cancel_right, \u2190 (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081] at h\u2081'\n rw [h\u2082.entropy_eq, add_sub_cancel_right, \u2190 (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082] at h\u2082'\n linarith", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 634}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\nopen ElementaryAddCommGroup\n\n/-- In fact $k$ is at most\n $$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\nlemma construct_good_improved' :\n k \u2264 \u03b4 + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 # T\u2081 | T\u2082] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2082 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2082 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2083 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2083 | T\u2082] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2081 | T\u2082] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2081 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2083 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2083 | T\u2082] - d[p.X\u2080\u2082 # X\u2082])) := by\n have I1 : I[T\u2082 : T\u2081] = I[T\u2081 : T\u2082] := mutualInfo_comm hT\u2082 hT\u2081 _\n have I2 : I[T\u2083 : T\u2081] = I[T\u2081 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2081 _\n have I3 : I[T\u2083 : T\u2082] = I[T\u2082 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2082 _\n have Z123 := construct_good_prelim' h_min hT hT\u2081 hT\u2082 hT\u2083\n have h132 : T\u2081 + T\u2083 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z132 := construct_good_prelim' h_min h132 hT\u2081 hT\u2083 hT\u2082\n have h213 : T\u2082 + T\u2081 + T\u2083 = 0 := by rw [\u2190 hT]; abel\n have Z213 := construct_good_prelim' h_min h213 hT\u2082 hT\u2081 hT\u2083\n have h231 : T\u2082 + T\u2083 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z231 := construct_good_prelim' h_min h231 hT\u2082 hT\u2083 hT\u2081\n have h312 : T\u2083 + T\u2081 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z312 := construct_good_prelim' h_min h312 hT\u2083 hT\u2081 hT\u2082\n have h321 : T\u2083 + T\u2082 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z321 := construct_good_prelim' h_min h321 hT\u2083 hT\u2082 hT\u2081\n simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321\n linarith\n\n/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\nlemma construct_good_improved'' {\u03a9' : Type*} [MeasurableSpace \u03a9'] (\u03bc : Measure \u03a9')\n [IsProbabilityMeasure \u03bc] {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G}\n (hT : T\u2081 + T\u2082 + T\u2083 = 0) (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083) :\n k \u2264 I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc] + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]))\n := by\n let M : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n exact construct_good_improved' h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend aux\n\nopen scoped BigOperators\n\n/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\nlemma averaged_construct_good : k \u2264 (I[U : V | S] + I[V : W | S] + I[W : U | S])\n + (p.\u03b7 / 6) * (((d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]))\n + ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])))\n := by\n have hS : Measurable S := ((hX\u2081.add' hX\u2082).add' hX\u2081').add' hX\u2082'\n have hU : Measurable U := hX\u2081.add' hX\u2082\n have hV : Measurable V := hX\u2081'.add hX\u2082\n have hW : Measurable W := hX\u2081'.add hX\u2081\n have hUVW : U + V + W = 0 := sum_uvw_eq_zero X\u2081 X\u2082 X\u2081'\n have hz (a : \u211d) : a = \u2211 z, (\u2119 (S \u207b\u00b9' {z})).toReal * a := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u2119 hS, one_mul]\n rw [hz k, hz (d[p.X\u2080\u2081 # X\u2081]), hz (d[p.X\u2080\u2082 # X\u2082])]\n simp only [condMutualInfo_eq_sum' hS, \u2190 Finset.sum_add_distrib, \u2190 mul_add,\n condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, \u2190 Finset.sum_sub_distrib, \u2190 mul_sub, Finset.mul_sum,\n \u2190 mul_assoc (p.\u03b7/6), mul_comm (p.\u03b7/6), mul_assoc _ _ (p.\u03b7/6)]\n rw [Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_le_sum (fun i _hi \u21a6 ?_)\n rcases eq_or_ne (\u2119 (S \u207b\u00b9' {i})) 0 with h'i|h'i\n \u00b7 simp [h'i]\n rw [mul_assoc, \u2190 mul_add]\n gcongr\n have : IsProbabilityMeasure (\u2119[|S \u207b\u00b9' {i}]) := cond_isProbabilityMeasure \u2119 h'i\n linarith [construct_good_improved'' h_min (\u2119[|S \u207b\u00b9' {i}]) hUVW hU hV hW]\n\nvariable (p)\n\nlemma dist_diff_bound_1 :\n (d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n \u2264 (16 * k + 6 * d[X\u2081 # X\u2081] + 2 * d[X\u2082 # X\u2082]) / 4 + (H[X\u2081 + X\u2081'] - H[X\u2082 + X\u2082']) / 4\n + (H[X\u2082 | X\u2082 + X\u2082'] - H[X\u2081 | X\u2081 + X\u2081']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2082' X\u2081' hX\u2081 hX\u2082 hX\u2082' hX\u2081' h_indep.reindex_four_abcd\n have I2 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep.reindex_four_abdc\n have I3 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2082 X\u2081' hX\u2081 hX\u2082' hX\u2082 hX\u2081' h_indep.reindex_four_acbd\n have I4 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2081' X\u2082 hX\u2081 hX\u2082' hX\u2081' hX\u2082 h_indep.reindex_four_acdb\n have I5 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082 X\u2082' hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_adbc\n have I6 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082' X\u2082 hX\u2081 hX\u2081' hX\u2082' hX\u2082 h_indep.reindex_four_adcb\n have C1 : U + X\u2082' + X\u2081' = S := by abel\n have C2 : W + X\u2082 + X\u2082' = S := by abel\n have C3 : X\u2081 + X\u2082' + X\u2082 + X\u2081' = S := by abel\n have C4 : X\u2081 + X\u2082' + X\u2081' + X\u2082 = S := by abel\n have C5 : W + X\u2082' + X\u2082 = S := by abel\n have C7 : X\u2082 + X\u2081' = V := by abel\n have C8 : X\u2081 + X\u2081' = W := by abel\n have C9 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082.symm\n have C10 : d[X\u2082 # X\u2081'] = d[X\u2081' # X\u2082] := rdist_symm\n have C11 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081.symm\n have C12 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq h\u2082.symm\n have C13 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] := (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082.symm\n have C14 : d[X\u2081' # X\u2082] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C15 : H[X\u2081' + X\u2082'] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C16 : H[X\u2082'] = H[X\u2082] := h\u2082.symm.entropy_eq\n have C17 : H[X\u2081'] = H[X\u2081] := h\u2081.symm.entropy_eq\n have C18 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C19 : H[X\u2082' + X\u2081'] = H[U] := by rw [add_comm]; exact C15\n have C20 : d[X\u2082' # X\u2082] = d[X\u2082 # X\u2082] := h\u2082.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C21 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C22 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C23 : X\u2082' + X\u2082 = X\u2082 + X\u2082' := by abel\n have C24 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C25 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C27 : H[X\u2082' | X\u2082 + X\u2082'] = H[X\u2082 | X\u2082 + X\u2082'] := by\n conv_lhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2082'.add' hX\u2082) hX\u2082 (hX\u2082.add' hX\u2082')\n have I : IdentDistrib (\u27e8X\u2082', X\u2082\u27e9) (\u27e8X\u2082, X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2082.symm h\u2082\n (h_indep.indepFun (show 2 \u2260 1 by decide)) (h_indep.indepFun (show 1 \u2260 2 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C28 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C29 : H[X\u2081' | V] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082) hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C30 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n simp only [C1, C2, C3, C4, C5, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30] at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma dist_diff_bound_2 :\n ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]))\n \u2264 (16 * k + 6 * d[X\u2082 # X\u2082] + 2 * d[X\u2081 # X\u2081]) / 4 + (H[X\u2082 + X\u2082'] - H[X\u2081 + X\u2081']) / 4\n + (H[X\u2081 | X\u2081 + X\u2081'] - H[X\u2082 | X\u2082 + X\u2082']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2082' X\u2081' hX\u2082 hX\u2081 hX\u2082' hX\u2081' h_indep.reindex_four_bacd\n have I2 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2081' X\u2082' hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_badc\n have I3 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081 X\u2081' hX\u2082 hX\u2082' hX\u2081 hX\u2081' h_indep.reindex_four_bcad\n have I4 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081' X\u2081 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bcda\n have I5 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2081 X\u2082' hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bdac\n have I6 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2082' X\u2081 hX\u2082 hX\u2081' hX\u2082' hX\u2081 h_indep.reindex_four_bdca\n have C1 : X\u2082 + X\u2081 = X\u2081 + X\u2082 := by abel\n have C2 : X\u2081 + X\u2081' = W := by abel\n have C3 : U + X\u2082' + X\u2081' = S := by abel\n have C4 : X\u2082 + X\u2081' = V := by abel\n have C5 : X\u2082 + X\u2082' + X\u2081 + X\u2081' = S := by abel\n have C6 : X\u2082 + X\u2082' + X\u2081' + X\u2081 = S := by abel\n have C7 : V + X\u2081 + X\u2082' = S := by abel\n have C8 : V + X\u2082' + X\u2081 = S := by abel\n have C9 : d[X\u2082 # X\u2081] = d[X\u2081 # X\u2082] := rdist_symm\n have C10 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n have C11 : d[X\u2082 # X\u2081'] = d[X\u2081 # X\u2082] := by\n rw [rdist_symm]\n exact ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n have C12 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C13 : d[X\u2082' # X\u2081] = d[X\u2081 # X\u2082'] := rdist_symm\n have C14 : d[X\u2081' # X\u2081] = d[X\u2081 # X\u2081'] := rdist_symm\n have C15 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n have C16 : H[X\u2081' + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C17 : H[X\u2082' + X\u2081'] = H[X\u2081 + X\u2082] := by rw [add_comm]; exact C16\n have C18 : H[X\u2081'] = H[X\u2081] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2081.symm\n have C19 : H[X\u2082'] = H[X\u2082] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2082.symm\n have C20 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C21 : H[X\u2081' | W] = H[X\u2081 | W] := by\n conv_rhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2081) hX\u2081 (hX\u2081.add' hX\u2081')\n have I : IdentDistrib (\u27e8X\u2081', X\u2081\u27e9) (\u27e8X\u2081, X\u2081'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2081\n (h_indep.indepFun (show 3 \u2260 0 by decide)) (h_indep.indepFun (show 0 \u2260 3 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C22 : H[X\u2082' | X\u2082' + X\u2081] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C23 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C24 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C25 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C27 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n have C28 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C29 : H[X\u2082' + X\u2081] = H[X\u2081 + X\u2082] := by\n rw [add_comm]\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C30 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2081.symm\n have C31 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable) h\u2082.symm\n simp only [C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C25, C26, C27, C28, C29, C30, C31]\n at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma averaged_final : k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) := by\n apply (averaged_construct_good hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_min).trans\n have : 0 \u2264 p.\u03b7 := p.h\u03b7.le\n have := sum_condMutual_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep.reindex_four_abdc h_min\n gcongr ?_ + (p.\u03b7 / 6) * ?_\n linarith [dist_diff_bound_1 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep,\n dist_diff_bound_2 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep]\n\n/-- Suppose $0 < \\eta < 1/8$. Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$. The proof\nof this lemma uses copies `X\u2081', X\u2082'` already in the context. For a version that does not assume\nthese are given and constructs them instead, use `tau_strictly_decreases'`.\n-/\ntheorem tau_strictly_decreases_aux' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 := by\n have : 0 < p.\u03b7 := p.h\u03b7\n have : k \u2264 8 * p.\u03b7 * k := by calc\n k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) :=\n averaged_final p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n _ \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) +\n p.\u03b7 / 6 * (8 * k + 2 * (2 * (k + (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7)))) := by\n gcongr\n exact second_estimate_aux p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082\n h_indep.reindex_four_abdc h_min\n _ = 8 * p.\u03b7 * k - (1 - 5 * p.\u03b7 - 4/6 * p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7) := by ring\n _ \u2264 8 * p.\u03b7 * k := by\n simp only [tsub_le_iff_right, le_add_iff_nonneg_right]\n apply div_nonneg _ (by linarith)\n apply mul_nonneg (by linarith) _\n linarith [first_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min]\n apply le_antisymm _ (rdist_nonneg hX\u2081 hX\u2082)\n nlinarith\n\ntheorem tau_strictly_decreases' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 := by\n let \u27e8A, mA, \u03bc, Y\u2081, Y\u2082, Y\u2081', Y\u2082', h\u03bc, h_indep, hY\u2081, hY\u2082, hY\u2081', hY\u2082', h_id1, h_id2, h_id1', h_id2'\u27e9\n := independent_copies4_nondep hX\u2081 hX\u2082 hX\u2081 hX\u2082 \u2119 \u2119 \u2119 \u2119\n rw [\u2190 h_id1.rdist_eq h_id2]\n letI : MeasureSpace A := \u27e8\u03bc\u27e9\n have : IsProbabilityMeasure (\u2119 : Measure A) := h\u03bc\n rw [\u2190 h_id1.tau_minimizes p h_id2] at h_min\n exact tau_strictly_decreases_aux' p hY\u2081 hY\u2082 hY\u2081' hY\u2082' (h_id1.trans h_id1'.symm)\n (h_id2.trans h_id2'.symm) h_indep.reindex_four_abdc h_min hp\n\nend MainEstimates\n\nsection EntropicPFR\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nopen scoped Topology\nopen Filter Set\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n [MeasurableSingletonClass G]\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\n/-- For `p.\u03b7 \u2264 1/8`, there exist \u03c4-minimizers `X\u2081, X\u2082` at zero Rusza distance. For `p.\u03b7 < 1/8`,\nall minimizers are fine, by `tau_strictly_decreases'`. For `p.\u03b7 = 1/8`, we use a limit of\nminimizers for `\u03b7 < 1/8`, which exists by compactness. -/\nlemma tau_minimizer_exists_rdist_eq_zero :\n \u2203 (\u03a9 : Type uG) (m\u03a9 : MeasureSpace \u03a9) (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G),\n Measurable X\u2081 \u2227 Measurable X\u2082 \u2227 IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 tau_minimizes p X\u2081 X\u2082\n \u2227 d[X\u2081 # X\u2082] = 0 := by\n -- let `u\u2099` be a sequence converging from below to `\u03b7`. In particular, `u\u2099 < 1/8`.\n obtain \u27e8u, -, u_mem, u_lim\u27e9 :\n \u2203 u, StrictMono u \u2227 (\u2200 (n : \u2115), u n \u2208 Set.Ioo 0 p.\u03b7) \u2227 Tendsto u atTop (\ud835\udcdd p.\u03b7) :=\n exists_seq_strictMono_tendsto' p.h\u03b7\n -- For each `n`, consider a minimizer associated to `\u03b7 = u\u2099`.\n let q : \u2115 \u2192 refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G := fun n \u21a6\n \u27e8p.X\u2080\u2081, p.X\u2080\u2082, p.hmeas1, p.hmeas2, u n, (u_mem n).1, by linarith [(u_mem n).2, p.h\u03b7']\u27e9\n have : \u2200 n, \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | q n] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | q n] :=\n fun n \u21a6 tau_min_exists_measure (q n)\n choose \u03bc \u03bc1_prob \u03bc2_prob h\u03bc using this\n -- The minimizer associated to `u\u2099` is at zero Rusza distance of itself, by\n -- lemma `tau_strictly_decreases'`.\n have I n : d[id ; (\u03bc n).1 # id ; (\u03bc n).2] = 0 := by\n let M : MeasureSpace (G \u00d7 G) := \u27e8(\u03bc n).1.prod (\u03bc n).2\u27e9\n have : IsProbabilityMeasure ((\u03bc n).1.prod (\u03bc n).2) := by infer_instance\n have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; (\u03bc n).1 # id ; (\u03bc n).2] :=\n IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id\n rw [\u2190 this]\n apply tau_strictly_decreases' (q n) measurable_fst measurable_snd ?_\n (by linarith [(u_mem n).2, p.h\u03b7'])\n intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | q n] = \u03c4[id ; (\u03bc n).1 # id ; (\u03bc n).2 | q n] :=\n ProbabilityTheory.IdentDistrib.tau_eq (q n) IdentDistrib.fst_id IdentDistrib.snd_id\n rw [A]\n exact h\u03bc n _ _ h\u2081 h\u2082\n -- extract a converging subsequence of the sequence of minimizers, seen as pairs of probability\n -- measures on `G` (which is a compact space).\n let \u03bc' : \u2115 \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun n \u21a6 (\u27e8(\u03bc n).1, \u03bc1_prob n\u27e9, \u27e8(\u03bc n).2, \u03bc2_prob n\u27e9)\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G)\n have : DiscreteTopology G := \u27e8rfl\u27e9\n -- The limiting pair of measures will be the desired minimizer.\n rcases IsCompact.tendsto_subseq (x := \u03bc') isCompact_univ (fun n \u21a6 mem_univ _)\n with \u27e8\u03bd, -, \u03c6, \u03c6mono, h\u03bd\u27e9\n have \u03c6lim : Tendsto \u03c6 atTop atTop := \u03c6mono.tendsto_atTop\n let M : MeasureSpace (G \u00d7 G) := \u27e8(\u03bd.1 : Measure G).prod \u03bd.2\u27e9\n have P : IsProbabilityMeasure ((\u03bd.1 : Measure G).prod (\u03bd.2 : Measure G)) := by infer_instance\n refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_, ?_\u27e9\n -- check that it is indeed a minimizer, as a limit of minimizers.\n \u00b7 intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bd.1 # id ; \u03bd.2 | p] :=\n ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n rw [A]\n have L1 : Tendsto (fun n \u21a6 \u03c4[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2 | q (\u03c6 n)]) atTop\n (\ud835\udcdd (\u03c4[id ; \u03bd.1 # id ; \u03bd.2 | p])) := by\n apply Tendsto.add (Tendsto.add ?_ (Tendsto.mul (u_lim.comp \u03c6lim) ?_))\n (Tendsto.mul (u_lim.comp \u03c6lim) ?_)\n \u00b7 apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) h\u03bd\n \u00b7 have : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n apply Tendsto.comp (this.tendsto _) h\u03bd\n \u00b7 have : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n apply Tendsto.comp (this.tendsto _) h\u03bd\n have L2 : Tendsto (fun n \u21a6 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | q (\u03c6 n)]) atTop\n (\ud835\udcdd (\u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p])) :=\n Tendsto.add (Tendsto.add tendsto_const_nhds (Tendsto.mul (u_lim.comp \u03c6lim)\n tendsto_const_nhds)) (Tendsto.mul (u_lim.comp \u03c6lim) tendsto_const_nhds)\n exact le_of_tendsto_of_tendsto' L1 L2 (fun n \u21a6 h\u03bc (\u03c6 n) _ _ h\u2081 h\u2082)\n -- check that it has zero Rusza distance, as a limit of a sequence at zero Rusza distance.\n \u00b7 have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; \u03bd.1 # id ; \u03bd.2] :=\n IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id\n rw [this]\n have L1 : Tendsto (fun n \u21a6 d[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2]) atTop\n (\ud835\udcdd (d[id ; \u03bd.1 # id ; (\u03bd.2 : Measure G)])) := by\n apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) h\u03bd\n have L2 : Tendsto (fun n \u21a6 d[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2]) atTop (\ud835\udcdd 0) := by simp [I]\n exact tendsto_nhds_unique L1 L2\n\n", "theoremStatement": "/-- `entropic_PFR_conjecture_improv`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\n subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture_improv (hp\u03b7 : p.\u03b7 = 1/8) :\n \u2203 H : AddSubgroup G, \u2203 \u03a9 : Type uG, \u2203 m\u03a9 : MeasureSpace \u03a9, \u2203 U : \u03a9 \u2192 G,\n IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 Measurable U \u2227\n IsUniform H U \u2227 d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 10 * d[p.X\u2080\u2081 # 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"Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8\u03a9', m\u03a9', X\u2081, X\u2082, hX\u2081, hX\u2082, hP, htau_min, hdist\u27e9 := tau_minimizer_exists_rdist_eq_zero p\n obtain \u27e8H, U, hU, hH_unif, hdistX\u2081, hdistX\u2082\u27e9 := exists_isUniform_of_rdist_eq_zero hX\u2081 hX\u2082 hdist\n refine \u27e8H, \u03a9', inferInstance, U, inferInstance, hU, hH_unif , ?_\u27e9\n have h : \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[p.X\u2080\u2082 # p.X\u2080\u2081 | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1\n rw [tau, tau, hp\u03b7] at h\n norm_num at h\n have : d[p.X\u2080\u2081 # p.X\u2080\u2082] = d[p.X\u2080\u2082 # p.X\u2080\u2081] := rdist_symm\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # X\u2081] + d[X\u2081 # U] := rdist_triangle p.hmeas1 hX\u2081 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # X\u2082] + d[X\u2082 # U] := rdist_triangle p.hmeas2 hX\u2082 hU\n linarith", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 641}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\nopen ElementaryAddCommGroup\n\n/-- In fact $k$ is at most\n $$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\nlemma construct_good_improved' :\n k \u2264 \u03b4 + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 # T\u2081 | T\u2082] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2082 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2082 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2083 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2083 | T\u2082] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2081 | T\u2082] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2081 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2083 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2083 | T\u2082] - d[p.X\u2080\u2082 # X\u2082])) := by\n have I1 : I[T\u2082 : T\u2081] = I[T\u2081 : T\u2082] := mutualInfo_comm hT\u2082 hT\u2081 _\n have I2 : I[T\u2083 : T\u2081] = I[T\u2081 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2081 _\n have I3 : I[T\u2083 : T\u2082] = I[T\u2082 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2082 _\n have Z123 := construct_good_prelim' h_min hT hT\u2081 hT\u2082 hT\u2083\n have h132 : T\u2081 + T\u2083 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z132 := construct_good_prelim' h_min h132 hT\u2081 hT\u2083 hT\u2082\n have h213 : T\u2082 + T\u2081 + T\u2083 = 0 := by rw [\u2190 hT]; abel\n have Z213 := construct_good_prelim' h_min h213 hT\u2082 hT\u2081 hT\u2083\n have h231 : T\u2082 + T\u2083 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z231 := construct_good_prelim' h_min h231 hT\u2082 hT\u2083 hT\u2081\n have h312 : T\u2083 + T\u2081 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z312 := construct_good_prelim' h_min h312 hT\u2083 hT\u2081 hT\u2082\n have h321 : T\u2083 + T\u2082 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z321 := construct_good_prelim' h_min h321 hT\u2083 hT\u2082 hT\u2081\n simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321\n linarith\n\n/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\nlemma construct_good_improved'' {\u03a9' : Type*} [MeasurableSpace \u03a9'] (\u03bc : Measure \u03a9')\n [IsProbabilityMeasure \u03bc] {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G}\n (hT : T\u2081 + T\u2082 + T\u2083 = 0) (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083) :\n k \u2264 I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc] + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]))\n := by\n let M : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n exact construct_good_improved' h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend aux\n\nopen scoped BigOperators\n\n/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\nlemma averaged_construct_good : k \u2264 (I[U : V | S] + I[V : W | S] + I[W : U | S])\n + (p.\u03b7 / 6) * (((d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]))\n + ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])))\n := by\n have hS : Measurable S := ((hX\u2081.add' hX\u2082).add' hX\u2081').add' hX\u2082'\n have hU : Measurable U := hX\u2081.add' hX\u2082\n have hV : Measurable V := hX\u2081'.add hX\u2082\n have hW : Measurable W := hX\u2081'.add hX\u2081\n have hUVW : U + V + W = 0 := sum_uvw_eq_zero X\u2081 X\u2082 X\u2081'\n have hz (a : \u211d) : a = \u2211 z, (\u2119 (S \u207b\u00b9' {z})).toReal * a := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u2119 hS, one_mul]\n rw [hz k, hz (d[p.X\u2080\u2081 # X\u2081]), hz (d[p.X\u2080\u2082 # X\u2082])]\n simp only [condMutualInfo_eq_sum' hS, \u2190 Finset.sum_add_distrib, \u2190 mul_add,\n condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, \u2190 Finset.sum_sub_distrib, \u2190 mul_sub, Finset.mul_sum,\n \u2190 mul_assoc (p.\u03b7/6), mul_comm (p.\u03b7/6), mul_assoc _ _ (p.\u03b7/6)]\n rw [Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_le_sum (fun i _hi \u21a6 ?_)\n rcases eq_or_ne (\u2119 (S \u207b\u00b9' {i})) 0 with h'i|h'i\n \u00b7 simp [h'i]\n rw [mul_assoc, \u2190 mul_add]\n gcongr\n have : IsProbabilityMeasure (\u2119[|S \u207b\u00b9' {i}]) := cond_isProbabilityMeasure \u2119 h'i\n linarith [construct_good_improved'' h_min (\u2119[|S \u207b\u00b9' {i}]) hUVW hU hV hW]\n\nvariable (p)\n\nlemma dist_diff_bound_1 :\n (d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n \u2264 (16 * k + 6 * d[X\u2081 # X\u2081] + 2 * d[X\u2082 # X\u2082]) / 4 + (H[X\u2081 + X\u2081'] - H[X\u2082 + X\u2082']) / 4\n + (H[X\u2082 | X\u2082 + X\u2082'] - H[X\u2081 | X\u2081 + X\u2081']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2082' X\u2081' hX\u2081 hX\u2082 hX\u2082' hX\u2081' h_indep.reindex_four_abcd\n have I2 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep.reindex_four_abdc\n have I3 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2082 X\u2081' hX\u2081 hX\u2082' hX\u2082 hX\u2081' h_indep.reindex_four_acbd\n have I4 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2081' X\u2082 hX\u2081 hX\u2082' hX\u2081' hX\u2082 h_indep.reindex_four_acdb\n have I5 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082 X\u2082' hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_adbc\n have I6 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082' X\u2082 hX\u2081 hX\u2081' hX\u2082' hX\u2082 h_indep.reindex_four_adcb\n have C1 : U + X\u2082' + X\u2081' = S := by abel\n have C2 : W + X\u2082 + X\u2082' = S := by abel\n have C3 : X\u2081 + X\u2082' + X\u2082 + X\u2081' = S := by abel\n have C4 : X\u2081 + X\u2082' + X\u2081' + X\u2082 = S := by abel\n have C5 : W + X\u2082' + X\u2082 = S := by abel\n have C7 : X\u2082 + X\u2081' = V := by abel\n have C8 : X\u2081 + X\u2081' = W := by abel\n have C9 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082.symm\n have C10 : d[X\u2082 # X\u2081'] = d[X\u2081' # X\u2082] := rdist_symm\n have C11 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081.symm\n have C12 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq h\u2082.symm\n have C13 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] := (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082.symm\n have C14 : d[X\u2081' # X\u2082] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C15 : H[X\u2081' + X\u2082'] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C16 : H[X\u2082'] = H[X\u2082] := h\u2082.symm.entropy_eq\n have C17 : H[X\u2081'] = H[X\u2081] := h\u2081.symm.entropy_eq\n have C18 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C19 : H[X\u2082' + X\u2081'] = H[U] := by rw [add_comm]; exact C15\n have C20 : d[X\u2082' # X\u2082] = d[X\u2082 # X\u2082] := h\u2082.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C21 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C22 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C23 : X\u2082' + X\u2082 = X\u2082 + X\u2082' := by abel\n have C24 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C25 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C27 : H[X\u2082' | X\u2082 + X\u2082'] = H[X\u2082 | X\u2082 + X\u2082'] := by\n conv_lhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2082'.add' hX\u2082) hX\u2082 (hX\u2082.add' hX\u2082')\n have I : IdentDistrib (\u27e8X\u2082', X\u2082\u27e9) (\u27e8X\u2082, X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2082.symm h\u2082\n (h_indep.indepFun (show 2 \u2260 1 by decide)) (h_indep.indepFun (show 1 \u2260 2 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C28 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C29 : H[X\u2081' | V] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082) hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C30 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n simp only [C1, C2, C3, C4, C5, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30] at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma dist_diff_bound_2 :\n ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]))\n \u2264 (16 * k + 6 * d[X\u2082 # X\u2082] + 2 * d[X\u2081 # X\u2081]) / 4 + (H[X\u2082 + X\u2082'] - H[X\u2081 + X\u2081']) / 4\n + (H[X\u2081 | X\u2081 + X\u2081'] - H[X\u2082 | X\u2082 + X\u2082']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2082' X\u2081' hX\u2082 hX\u2081 hX\u2082' hX\u2081' h_indep.reindex_four_bacd\n have I2 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2081' X\u2082' hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_badc\n have I3 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081 X\u2081' hX\u2082 hX\u2082' hX\u2081 hX\u2081' h_indep.reindex_four_bcad\n have I4 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081' X\u2081 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bcda\n have I5 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2081 X\u2082' hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bdac\n have I6 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2082' X\u2081 hX\u2082 hX\u2081' hX\u2082' hX\u2081 h_indep.reindex_four_bdca\n have C1 : X\u2082 + X\u2081 = X\u2081 + X\u2082 := by abel\n have C2 : X\u2081 + X\u2081' = W := by abel\n have C3 : U + X\u2082' + X\u2081' = S := by abel\n have C4 : X\u2082 + X\u2081' = V := by abel\n have C5 : X\u2082 + X\u2082' + X\u2081 + X\u2081' = S := by abel\n have C6 : X\u2082 + X\u2082' + X\u2081' + X\u2081 = S := by abel\n have C7 : V + X\u2081 + X\u2082' = S := by abel\n have C8 : V + X\u2082' + X\u2081 = S := by abel\n have C9 : d[X\u2082 # X\u2081] = d[X\u2081 # X\u2082] := rdist_symm\n have C10 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n have C11 : d[X\u2082 # X\u2081'] = d[X\u2081 # X\u2082] := by\n rw [rdist_symm]\n exact ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n have C12 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C13 : d[X\u2082' # X\u2081] = d[X\u2081 # X\u2082'] := rdist_symm\n have C14 : d[X\u2081' # X\u2081] = d[X\u2081 # X\u2081'] := rdist_symm\n have C15 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n have C16 : H[X\u2081' + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C17 : H[X\u2082' + X\u2081'] = H[X\u2081 + X\u2082] := by rw [add_comm]; exact C16\n have C18 : H[X\u2081'] = H[X\u2081] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2081.symm\n have C19 : H[X\u2082'] = H[X\u2082] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2082.symm\n have C20 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C21 : H[X\u2081' | W] = H[X\u2081 | W] := by\n conv_rhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2081) hX\u2081 (hX\u2081.add' hX\u2081')\n have I : IdentDistrib (\u27e8X\u2081', X\u2081\u27e9) (\u27e8X\u2081, X\u2081'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2081\n (h_indep.indepFun (show 3 \u2260 0 by decide)) (h_indep.indepFun (show 0 \u2260 3 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C22 : H[X\u2082' | X\u2082' + X\u2081] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C23 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C24 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C25 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C27 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n have C28 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C29 : H[X\u2082' + X\u2081] = H[X\u2081 + X\u2082] := by\n rw [add_comm]\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C30 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2081.symm\n have C31 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable) h\u2082.symm\n simp only [C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C25, C26, C27, C28, C29, C30, C31]\n at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma averaged_final : k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) := by\n apply (averaged_construct_good hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_min).trans\n have : 0 \u2264 p.\u03b7 := p.h\u03b7.le\n have := sum_condMutual_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep.reindex_four_abdc h_min\n gcongr ?_ + (p.\u03b7 / 6) * ?_\n linarith [dist_diff_bound_1 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep,\n dist_diff_bound_2 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep]\n\n/-- Suppose $0 < \\eta < 1/8$. Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$. The proof\nof this lemma uses copies `X\u2081', X\u2082'` already in the context. For a version that does not assume\nthese are given and constructs them instead, use `tau_strictly_decreases'`.\n-/\ntheorem tau_strictly_decreases_aux' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 := by\n have : 0 < p.\u03b7 := p.h\u03b7\n have : k \u2264 8 * p.\u03b7 * k := by calc\n k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) :=\n averaged_final p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n _ \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) +\n p.\u03b7 / 6 * (8 * k + 2 * (2 * (k + (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7)))) := by\n gcongr\n exact second_estimate_aux p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082\n h_indep.reindex_four_abdc h_min\n _ = 8 * p.\u03b7 * k - (1 - 5 * p.\u03b7 - 4/6 * p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7) := by ring\n _ \u2264 8 * p.\u03b7 * k := by\n simp only [tsub_le_iff_right, le_add_iff_nonneg_right]\n apply div_nonneg _ (by linarith)\n apply mul_nonneg (by linarith) _\n linarith [first_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min]\n apply le_antisymm _ (rdist_nonneg hX\u2081 hX\u2082)\n nlinarith\n\ntheorem tau_strictly_decreases' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 := by\n let \u27e8A, mA, \u03bc, Y\u2081, Y\u2082, Y\u2081', Y\u2082', h\u03bc, h_indep, hY\u2081, hY\u2082, hY\u2081', hY\u2082', h_id1, h_id2, h_id1', h_id2'\u27e9\n := independent_copies4_nondep hX\u2081 hX\u2082 hX\u2081 hX\u2082 \u2119 \u2119 \u2119 \u2119\n rw [\u2190 h_id1.rdist_eq h_id2]\n letI : MeasureSpace A := \u27e8\u03bc\u27e9\n have : IsProbabilityMeasure (\u2119 : Measure A) := h\u03bc\n rw [\u2190 h_id1.tau_minimizes p h_id2] at h_min\n exact tau_strictly_decreases_aux' p hY\u2081 hY\u2082 hY\u2081' hY\u2082' (h_id1.trans h_id1'.symm)\n (h_id2.trans h_id2'.symm) h_indep.reindex_four_abdc h_min hp\n\nend MainEstimates\n\nsection EntropicPFR\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nopen scoped Topology\nopen Filter Set\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n [MeasurableSingletonClass G]\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\n/-- For `p.\u03b7 \u2264 1/8`, there exist \u03c4-minimizers `X\u2081, X\u2082` at zero Rusza distance. For `p.\u03b7 < 1/8`,\nall minimizers are fine, by `tau_strictly_decreases'`. For `p.\u03b7 = 1/8`, we use a limit of\nminimizers for `\u03b7 < 1/8`, which exists by compactness. -/\nlemma tau_minimizer_exists_rdist_eq_zero :\n \u2203 (\u03a9 : Type uG) (m\u03a9 : MeasureSpace \u03a9) (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G),\n Measurable X\u2081 \u2227 Measurable X\u2082 \u2227 IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 tau_minimizes p X\u2081 X\u2082\n \u2227 d[X\u2081 # X\u2082] = 0 := by\n -- let `u\u2099` be a sequence converging from below to `\u03b7`. In particular, `u\u2099 < 1/8`.\n obtain \u27e8u, -, u_mem, u_lim\u27e9 :\n \u2203 u, StrictMono u \u2227 (\u2200 (n : \u2115), u n \u2208 Set.Ioo 0 p.\u03b7) \u2227 Tendsto u atTop (\ud835\udcdd p.\u03b7) :=\n exists_seq_strictMono_tendsto' p.h\u03b7\n -- For each `n`, consider a minimizer associated to `\u03b7 = u\u2099`.\n let q : \u2115 \u2192 refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G := fun n \u21a6\n \u27e8p.X\u2080\u2081, p.X\u2080\u2082, p.hmeas1, p.hmeas2, u n, (u_mem n).1, by linarith [(u_mem n).2, p.h\u03b7']\u27e9\n have : \u2200 n, \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | q n] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | q n] :=\n fun n \u21a6 tau_min_exists_measure (q n)\n choose \u03bc \u03bc1_prob \u03bc2_prob h\u03bc using this\n -- The minimizer associated to `u\u2099` is at zero Rusza distance of itself, by\n -- lemma `tau_strictly_decreases'`.\n have I n : d[id ; (\u03bc n).1 # id ; (\u03bc n).2] = 0 := by\n let M : MeasureSpace (G \u00d7 G) := \u27e8(\u03bc n).1.prod (\u03bc n).2\u27e9\n have : IsProbabilityMeasure ((\u03bc n).1.prod (\u03bc n).2) := by infer_instance\n have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; (\u03bc n).1 # id ; (\u03bc n).2] :=\n IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id\n rw [\u2190 this]\n apply tau_strictly_decreases' (q n) measurable_fst measurable_snd ?_\n (by linarith [(u_mem n).2, p.h\u03b7'])\n intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | q n] = \u03c4[id ; (\u03bc n).1 # id ; (\u03bc n).2 | q n] :=\n ProbabilityTheory.IdentDistrib.tau_eq (q n) IdentDistrib.fst_id IdentDistrib.snd_id\n rw [A]\n exact h\u03bc n _ _ h\u2081 h\u2082\n -- extract a converging subsequence of the sequence of minimizers, seen as pairs of probability\n -- measures on `G` (which is a compact space).\n let \u03bc' : \u2115 \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun n \u21a6 (\u27e8(\u03bc n).1, \u03bc1_prob n\u27e9, \u27e8(\u03bc n).2, \u03bc2_prob n\u27e9)\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G)\n have : DiscreteTopology G := \u27e8rfl\u27e9\n -- The limiting pair of measures will be the desired minimizer.\n rcases IsCompact.tendsto_subseq (x := \u03bc') isCompact_univ (fun n \u21a6 mem_univ _)\n with \u27e8\u03bd, -, \u03c6, \u03c6mono, h\u03bd\u27e9\n have \u03c6lim : Tendsto \u03c6 atTop atTop := \u03c6mono.tendsto_atTop\n let M : MeasureSpace (G \u00d7 G) := \u27e8(\u03bd.1 : Measure G).prod \u03bd.2\u27e9\n have P : IsProbabilityMeasure ((\u03bd.1 : Measure G).prod (\u03bd.2 : Measure G)) := by infer_instance\n refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_, ?_\u27e9\n -- check that it is indeed a minimizer, as a limit of minimizers.\n \u00b7 intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bd.1 # id ; \u03bd.2 | p] :=\n ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n rw [A]\n have L1 : Tendsto (fun n \u21a6 \u03c4[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2 | q (\u03c6 n)]) atTop\n (\ud835\udcdd (\u03c4[id ; \u03bd.1 # id ; \u03bd.2 | p])) := by\n apply Tendsto.add (Tendsto.add ?_ (Tendsto.mul (u_lim.comp \u03c6lim) ?_))\n (Tendsto.mul (u_lim.comp \u03c6lim) ?_)\n \u00b7 apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) h\u03bd\n \u00b7 have : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n apply Tendsto.comp (this.tendsto _) h\u03bd\n \u00b7 have : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n apply Tendsto.comp (this.tendsto _) h\u03bd\n have L2 : Tendsto (fun n \u21a6 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | q (\u03c6 n)]) atTop\n (\ud835\udcdd (\u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p])) :=\n Tendsto.add (Tendsto.add tendsto_const_nhds (Tendsto.mul (u_lim.comp \u03c6lim)\n tendsto_const_nhds)) (Tendsto.mul (u_lim.comp \u03c6lim) tendsto_const_nhds)\n exact le_of_tendsto_of_tendsto' L1 L2 (fun n \u21a6 h\u03bc (\u03c6 n) _ _ h\u2081 h\u2082)\n -- check that it has zero Rusza distance, as a limit of a sequence at zero Rusza distance.\n \u00b7 have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; \u03bd.1 # id ; \u03bd.2] :=\n IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id\n rw [this]\n have L1 : Tendsto (fun n \u21a6 d[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2]) atTop\n (\ud835\udcdd (d[id ; \u03bd.1 # id ; (\u03bd.2 : Measure G)])) := by\n apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) h\u03bd\n have L2 : Tendsto (fun n \u21a6 d[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2]) atTop (\ud835\udcdd 0) := by simp [I]\n exact tendsto_nhds_unique L1 L2\n\n/-- `entropic_PFR_conjecture_improv`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\n subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture_improv (hp\u03b7 : p.\u03b7 = 1/8) :\n \u2203 H : AddSubgroup G, \u2203 \u03a9 : Type uG, \u2203 m\u03a9 : MeasureSpace \u03a9, \u2203 U : \u03a9 \u2192 G,\n IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 Measurable U \u2227\n IsUniform H U \u2227 d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 10 * d[p.X\u2080\u2081 # p.X\u2080\u2082] := by\n obtain \u27e8\u03a9', m\u03a9', X\u2081, X\u2082, hX\u2081, hX\u2082, hP, htau_min, hdist\u27e9 := tau_minimizer_exists_rdist_eq_zero p\n obtain \u27e8H, U, hU, hH_unif, hdistX\u2081, hdistX\u2082\u27e9 := exists_isUniform_of_rdist_eq_zero hX\u2081 hX\u2082 hdist\n refine \u27e8H, \u03a9', inferInstance, U, inferInstance, hU, hH_unif , ?_\u27e9\n have h : \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[p.X\u2080\u2082 # p.X\u2080\u2081 | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1\n rw [tau, tau, hp\u03b7] at h\n norm_num at h\n have : d[p.X\u2080\u2081 # p.X\u2080\u2082] = d[p.X\u2080\u2082 # p.X\u2080\u2081] := rdist_symm\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # X\u2081] + d[X\u2081 # U] := rdist_triangle p.hmeas1 hX\u2081 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # X\u2082] + d[X\u2082 # U] := rdist_triangle p.hmeas2 hX\u2082 hU\n linarith\n\n", "theoremStatement": "/-- `entropic_PFR_conjecture_improv'`: For two $G$-valued random variables $X^0_1, X^0_2$, there is\nsome subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$., and\nd[X^0_1; U_H] and d[X^0_2; U_H] are at most 5/2 * d[X^0_1;X^0_2] -/\ntheorem entropic_PFR_conjecture_improv' (hp\u03b7 : p.\u03b7 = 1/8) :\n \u2203 H : AddSubgroup G, \u2203 \u03a9 : Type uG, \u2203 m\u03a9 : MeasureSpace \u03a9, \u2203 U : \u03a9 \u2192 G,\n IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 Measurable U \u2227\n IsUniform H U \u2227 d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 10 * d[p.X\u2080\u2081 # p.X\u2080\u2082] \u2227 d[p.X\u2080\u2081 # U] \u2264 11/2 * d[p.X\u2080\u2081 # p.X\u2080\u2082] \u2227 d[p.X\u2080\u2082 # U] \u2264 11/2 * d[p.X\u2080\u2081 # p.X\u2080\u2082] ", "theoremName": "entropic_PFR_conjecture_improv'", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "898757a4a68787d9da498a48a4c13841d0ab456f", "date": "2024-01-09"}, "file": 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"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.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "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.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.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.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.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.Paths", "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", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "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", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "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", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "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", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8\u03a9', m\u03a9', X\u2081, X\u2082, hX\u2081, hX\u2082, hP, htau_min, hdist\u27e9 := tau_minimizer_exists_rdist_eq_zero p\n obtain \u27e8H, U, hU, hH_unif, hdistX\u2081, hdistX\u2082\u27e9 := exists_isUniform_of_rdist_eq_zero hX\u2081 hX\u2082 hdist\n have : d[p.X\u2080\u2081 # p.X\u2080\u2082] = d[p.X\u2080\u2082 # p.X\u2080\u2081] := rdist_symm\n have goal\u2081 : d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 10 * d[p.X\u2080\u2081 # p.X\u2080\u2082] := by\n have h : \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[p.X\u2080\u2082 # p.X\u2080\u2081 | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1\n rw [tau, tau, hp\u03b7] at h\n norm_num at h\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # X\u2081] + d[X\u2081 # U] := rdist_triangle p.hmeas1 hX\u2081 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # X\u2082] + d[X\u2082 # U] := rdist_triangle p.hmeas2 hX\u2082 hU\n linarith\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # p.X\u2080\u2082] + d[p.X\u2080\u2082 # U] := rdist_triangle p.hmeas1 p.hmeas2 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # p.X\u2080\u2081] + d[p.X\u2080\u2081 # U] := rdist_triangle p.hmeas2 p.hmeas1 hU\n refine \u27e8H, \u03a9', inferInstance, U, inferInstance, hU, hH_unif, goal\u2081, by linarith, by linarith\u27e9", "proofType": "tactic", "proofLengthLines": 13, "proofLengthTokens": 945}} +{"srcContext": "import Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport PFR.ForMathlib.Entropy.Measure\nimport PFR.Mathlib.MeasureTheory.Integral.Bochner\nimport PFR.Mathlib.MeasureTheory.Measure.Typeclasses\nimport PFR.Mathlib.Probability.Kernel.Disintegration\n\n/-!\n# Entropy of a kernel with respect to a measure\n\n## Main definitions\n\n* `kernel.entropy`: entropy of a kernel `\u03ba` with respect to a measure `\u03bc`,\n `\u03bc[fun t \u21a6 measureEntropy (\u03ba t)]`. We use the notation `Hk[\u03ba, \u03bc]`.\n\n## Main statements\n\n* `chain_rule`: `Hk[\u03ba, \u03bc] = Hk[fst \u03ba, \u03bc] + Hk[condKernel \u03ba, \u03bc \u2297\u2098 (fst \u03ba)]`.\n* `entropy_map_le`: data-processing inequality for the kernel entropy\n\n## Notations\n\n* `Hk[\u03ba, \u03bc] = kernel.entropy \u03ba \u03bc`\n\n-/\n\nopen Real MeasureTheory\n\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\n\nnamespace ProbabilityTheory.kernel\n\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S]\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U]\n [Countable V] [Nonempty V] [MeasurableSpace V] [MeasurableSingletonClass V]\n\n/-- Entropy of a kernel with respect to a measure. -/\nnoncomputable\ndef entropy (\u03ba : kernel T S) (\u03bc : Measure T) := \u03bc[fun y \u21a6 Hm[\u03ba y]]\n\n/-- Entropy of a kernel with respect to a measure. -/\nnotation3:100 \"Hk[\" \u03ba \" , \" \u03bc \"]\" => ProbabilityTheory.kernel.entropy \u03ba \u03bc\n\n@[simp]\nlemma entropy_zero_measure (\u03ba : kernel T S) : Hk[\u03ba, (0 : Measure T)] = 0 := by simp [entropy]\n\n@[simp]\nlemma entropy_zero_kernel (\u03bc : Measure T) : Hk[(0 : kernel T S), \u03bc] = 0 := by simp [entropy]\n\nlemma entropy_congr {\u03ba \u03b7 : kernel T S} (h : \u03ba =\u1d50[\u03bc] \u03b7) : Hk[\u03ba, \u03bc] = Hk[\u03b7, \u03bc] := by\n simp_rw [entropy]\n refine integral_congr_ae ?_\n filter_upwards [h] with x hx\n rw [hx]\n\nlemma entropy_nonneg (\u03ba : kernel T S) (\u03bc : Measure T) :\n 0 \u2264 Hk[\u03ba, \u03bc] := integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n@[simp]\nlemma entropy_const (\u03bd : Measure S) (\u03bc : Measure T) :\n Hk[kernel.const T \u03bd, \u03bc] = (\u03bc Set.univ).toReal * Hm[\u03bd] := by\n rw [entropy]\n simp only [const_apply, MeasureTheory.integral_const, smul_eq_mul, ne_eq]\n\n/-- Constant kernels with finite support, have finite kernel support. -/\nlemma finiteKernelSupport_of_const (\u03bd : Measure S) [FiniteSupport \u03bd] :\n FiniteKernelSupport (kernel.const T \u03bd) := by\n intro t\n use \u03bd.support\n simp [measure_compl_support \u03bd]\n\n/-- Composing a finitely supported measure with a finitely supported kernel gives a finitely supported kernel. -/\nlemma finiteSupport_of_compProd' {\u03bc : Measure T} [IsFiniteMeasure \u03bc] {\u03ba : kernel T S}\n [IsMarkovKernel \u03ba] [FiniteSupport \u03bc] (h\u03ba : FiniteKernelSupport \u03ba) :\n FiniteSupport (\u03bc \u2297\u2098 \u03ba) := by\n let A := \u03bc.support\n have hA := measure_compl_support \u03bc\n rcases (local_support_of_finiteKernelSupport h\u03ba A) with \u27e8B, hB\u27e9\n use A \u00d7\u02e2 B\n rw [Measure.compProd_apply (by measurability), lintegral_eq_sum' _ hA]\n apply Finset.sum_eq_zero\n intro t ht\n simp; left\n refine measure_mono_null ?_ (hB t ht)\n intro s\n simp; tauto\n\nlemma finiteSupport_of_compProd {\u03bc : Measure T} [IsFiniteMeasure \u03bc] {\u03ba : kernel T S}\n [IsMarkovKernel \u03ba] [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n FiniteSupport (\u03bc \u2297\u2098 \u03ba) := by\n rw [Measure.compProd_congr h\u03ba.ae_eq_mk]\n exact finiteSupport_of_compProd' h\u03ba.finiteKernelSupport_mk\n\nlemma aefiniteKernelSupport_condDistrib (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc]\n (hX: Measurable X) (hY: Measurable Y) [FiniteRange X] [FiniteRange Y] :\n AEFiniteKernelSupport (condDistrib X Y \u03bc) (\u03bc.map Y) := by\n filter_upwards [condDistrib_ae_eq hX hY \u03bc] with a ha\n rw [ha]\n exact finiteSupport_of_finiteRange.finite\n\nlemma entropy_le_log_card (\u03ba : kernel T S) (\u03bc : Measure T) [Fintype S] [IsProbabilityMeasure \u03bc] :\n Hk[\u03ba, \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 measureEntropy_nonneg _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 measureEntropy_le_log_card _)\n \u00b7 simp\n\nlemma entropy_eq_integral_sum (\u03ba : kernel T S) [IsMarkovKernel \u03ba] (\u03bc : Measure T) :\n Hk[\u03ba, \u03bc] = \u03bc[fun y \u21a6 \u2211' x, negMulLog (\u03ba y {x}).toReal] := by\n simp_rw [entropy, measureEntropy_of_isProbabilityMeasure]\n\n-- entropy_map_of_injective is a special case of this (see def of map)\nlemma entropy_snd_compProd_deterministic_of_injective (\u03ba : kernel T S) [IsMarkovKernel \u03ba]\n (\u03bc : Measure T) {f : T \u00d7 S \u2192 U}\n (hf : \u2200 t, Function.Injective (fun x \u21a6 f (t, x))) (hmes : Measurable f) :\n Hk[snd (\u03ba \u2297\u2096 deterministic f hmes), \u03bc] = Hk[\u03ba, \u03bc] := by\n have : \u2200 t, snd (\u03ba \u2297\u2096 deterministic f hmes) t\n = map \u03ba (fun x \u21a6 f (t, x)) (hmes.comp measurable_prod_mk_left) t := by\n intro t\n ext s hs\n rw [snd_apply' _ _ hs, compProd_deterministic_apply, map_apply' _ _ _ hs]\n \u00b7 congr\n \u00b7 exact measurable_snd hs\n simp_rw [entropy]\n congr with y\n convert measureEntropy_map_of_injective (\u03ba y) _ (hmes.comp measurable_prod_mk_left) (hf y)\n rw [this y, map_apply]\n congr\n\nlemma entropy_map_of_injective\n (\u03ba : kernel T S) (\u03bc : Measure T) {f : S \u2192 U} (hf : Function.Injective f) (hmes : Measurable f) :\n Hk[map \u03ba f hmes, \u03bc] = Hk[\u03ba, \u03bc] := by\n simp_rw [entropy, map_apply, measureEntropy_map_of_injective _ _ hmes hf]\n\nlemma entropy_map_swap (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) :\n Hk[map \u03ba Prod.swap measurable_swap, \u03bc] = Hk[\u03ba, \u03bc] := entropy_map_of_injective \u03ba \u03bc Prod.swap_injective _\n\nlemma entropy_swapRight (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) :\n Hk[swapRight \u03ba, \u03bc] = Hk[\u03ba, \u03bc] := by\n rw [swapRight, entropy_map_swap]\n\nlemma entropy_comap {T' : Type*} [MeasurableSpace T'] [MeasurableSingletonClass T']\n (\u03ba : kernel T S) (\u03bc : Measure T) (f : T' \u2192 T) (hf : MeasurableEmbedding f)\n (hf_range : Set.range f =\u1d50[\u03bc] Set.univ)\n [IsFiniteMeasure \u03bc] [IsFiniteMeasure (\u03bc.comap f)] (hf\u03bc : FiniteSupport (\u03bc.comap f)) :\n Hk[comap \u03ba f hf.measurable, \u03bc.comap f] = Hk[\u03ba, \u03bc] := by\n classical\n rcases hf\u03bc with \u27e8A, hA\u27e9\n have : \u03bc (Finset.image f A : Set T)\u1d9c = 0 := by\n rw [Finset.coe_image, Function.Injective.compl_image_eq hf.injective]\n simp\n constructor\n . rwa [\u2190 Measure.comap_apply f hf.injective hf.measurableSet_image']\n exact MeasurableSet.compl (Finset.measurableSet A)\n exact ae_eq_univ.mp hf_range\n simp_rw [entropy]\n simp_rw [integral_eq_sum' _ hA, integral_eq_sum' _ this, Measure.comap_apply f hf.injective hf.measurableSet_image' _\n (measurableSet_singleton _)]\n simp only [Set.image_singleton, smul_eq_mul]\n simp_rw [comap_apply]\n rw [\u2190 Finset.sum_image (f := fun x \u21a6 (\u03bc {x}).toReal * measureEntropy (\u03ba x)) (g := f)]\n intro x _ y _ hxy\n exact hf.injective hxy\n\nlemma FiniteSupport.comap_equiv {T' : Type*} [MeasurableSpace T'] [MeasurableSingletonClass T']\n {\u03bc : Measure T} (f : T' \u2243\u1d50 T) [FiniteSupport \u03bc] :\n FiniteSupport (\u03bc.comap f) := by\n classical\n let A := \u03bc.support\n have hA := measure_compl_support \u03bc\n refine \u27e8Finset.image f.symm A, ?_\u27e9\n rwa [Finset.coe_image, \u2190 Set.image_compl_eq (MeasurableEquiv.bijective f.symm),\n Measure.comap_apply f (MeasurableEquiv.injective f),MeasurableEquiv.image_symm,\n MeasurableEquiv.image_preimage]\n . exact fun _ \u21a6 (MeasurableEquiv.measurableSet_image f).mpr\n \u00b7 exact f.symm.measurableSet_image.mpr A.measurableSet.compl\n\nlemma entropy_comap_equiv {T' : Type*} [MeasurableSpace T'] [MeasurableSingletonClass T']\n (\u03ba : kernel T S) {\u03bc : Measure T} (f : T' \u2243\u1d50 T)\n [IsFiniteMeasure \u03bc] [FiniteSupport \u03bc] :\n Hk[comap \u03ba f f.measurable, \u03bc.comap f] = Hk[\u03ba, \u03bc] := by\n rw [entropy_comap]\n \u00b7 exact f.measurableEmbedding\n \u00b7 rw [\u2190 MeasurableEquiv.coe_toEquiv, Equiv.range_eq_univ]\n \u00b7 exact FiniteSupport.comap_equiv f\n\nlemma entropy_comap_swap\n {T' : Type*} [MeasurableSpace T'] [MeasurableSingletonClass T'] [Nonempty T']\n (\u03ba : kernel (T' \u00d7 T) S) {\u03bc : Measure (T' \u00d7 T)} [IsFiniteMeasure \u03bc] [FiniteSupport \u03bc] :\n Hk[comap \u03ba Prod.swap measurable_swap, \u03bc.comap Prod.swap] = Hk[\u03ba, \u03bc] :=\n entropy_comap_equiv \u03ba MeasurableEquiv.prodComm\n\n/-- Measurable equivalence with the product with the one-point space `Unit`.-/\ndef prodMKLeft_unit_equiv (T : Type*) [MeasurableSpace T] : (Unit \u00d7 T) \u2243\u1d50 T where\n toFun := fun p \u21a6 p.2\n invFun:= Prod.mk ()\n left_inv := by intro p; simp\n right_inv := by intro t; simp\n measurable_toFun := by simp; exact measurable_snd\n measurable_invFun := by simp; exact measurable_prod_mk_left\n\nlemma entropy_prodMkLeft_unit (\u03ba : kernel T S) {\u03bc : Measure T} [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] :\n Hk[prodMkLeft Unit \u03ba, \u03bc.map (Prod.mk ())] = Hk[\u03ba, \u03bc] := by\n convert entropy_comap_equiv \u03ba (prodMKLeft_unit_equiv T) (\u03bc := \u03bc)\n funext \u03bc\n rw [\u2190 MeasurableEquiv.map_symm]\n congr\n\n", "theoremStatement": "lemma entropy_compProd_aux [IsFiniteMeasure \u03bc] {\u03ba : kernel T S} [IsMarkovKernel \u03ba]\n {\u03b7 : kernel (T \u00d7 S) U} [IsMarkovKernel \u03b7] [FiniteSupport \u03bc] (h\u03ba : FiniteKernelSupport \u03ba)\n (h\u03b7 : FiniteKernelSupport \u03b7) :\n Hk[\u03ba \u2297\u2096 \u03b7, \u03bc] = Hk[\u03ba, \u03bc]\n + \u03bc[fun t \u21a6 Hk[comap \u03b7 (Prod.mk t) measurable_prod_mk_left, (\u03ba t)]] ", "theoremName": "ProbabilityTheory.kernel.entropy_compProd_aux", 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"Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let A := \u03bc.support\n have hA := measure_compl_support \u03bc\n have hsum (F : T \u2192 \u211d) : \u222b (t : T), F t \u2202\u03bc = \u2211 t in A, (\u03bc.real {t}) * (F t) := by\n rw [integral_eq_sum' _ hA]\n congr with t ht\n simp_rw [entropy, hsum, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_congr rfl\n intro t ht\n rw [\u2190 mul_add]\n congr\n rcases (local_support_of_finiteKernelSupport h\u03ba A) with \u27e8B, hB\u27e9\n rcases (local_support_of_finiteKernelSupport h\u03b7 (A \u00d7\u02e2 B)) with \u27e8C, hC\u27e9\n rw [integral_eq_sum' _ (hB t ht)]\n have h\u03ba\u03b7 : ((\u03ba \u2297\u2096 \u03b7) t) (B \u00d7\u02e2 C: Finset (S \u00d7 U))\u1d9c = 0 := by\n rw [ProbabilityTheory.kernel.compProd_apply, lintegral_eq_sum' _ (hB t ht)]\n . apply Finset.sum_eq_zero\n intro s hs\n simp; left\n have hts : (t, s) \u2208 A \u00d7\u02e2 B := by simp [ht, hs]\n refine measure_mono_null ?_ (hC (t, s) hts)\n intro u hu\n simp at hu \u22a2\n exact hu hs\n exact MeasurableSet.compl (Finset.measurableSet _)\n rw [measureEntropy_def_finite' h\u03ba\u03b7, measureEntropy_def_finite' (hB t ht), \u2190 Finset.sum_add_distrib, Finset.sum_product]\n apply Finset.sum_congr rfl\n intro s hs\n simp\n have hts : (t, s) \u2208 A \u00d7\u02e2 B := by simp [ht, hs]\n have h\u03b7': (comap \u03b7 (Prod.mk t) measurable_prod_mk_left) s C\u1d9c = 0 := by\n rw [kernel.comap_apply]\n exact hC (t, s) hts\n rw [measureEntropy_def_finite' h\u03b7']\n simp\n have : negMulLog ((\u03ba t).real {s}) = \u2211 u in C, negMulLog ((\u03ba t).real {s}) * ((comap \u03b7 (Prod.mk t) measurable_prod_mk_left) s).real {u} := by\n rw [\u2190 Finset.mul_sum]\n simp\n rw [kernel.comap_apply]\n suffices (\u03b7 (t, s)).real \u2191C = (\u03b7 (t, s)).real Set.univ by simp [this]\n have := hC (t, s) hts\n rw [\u2190 measureReal_eq_zero_iff] at this\n rw [\u2190 measureReal_add_measureReal_compl (s := C) _, this, add_zero]\n exact Finset.measurableSet C\n rw [this, Finset.mul_sum, \u2190 Finset.sum_add_distrib]\n congr with u\n have : ((\u03ba \u2297\u2096 \u03b7) t).real {(s, u)} = ((\u03ba t).real {s}) * ((\u03b7 (t, s)).real {u}) := by\n rw [measureReal_def, compProd_apply \u03ba \u03b7 _ (measurableSet_singleton _), lintegral_eq_sum' _ (hB t ht), Finset.sum_eq_single_of_mem s hs]\n . simp [measureReal_def]; ring\n intro b _ hbs\n simp [hbs]\n rw [this, kernel.comap_apply, negMulLog_mul, negMulLog, negMulLog, \u2190 measureReal_def]\n ring", "proofType": "tactic", "proofLengthLines": 52, "proofLengthTokens": 2210}} +{"srcContext": "import PFR.FirstEstimate\nimport PFR.Mathlib.Probability.Independence.FourVariables\n\n/-!\n# Second estimate\n\nThe second estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ be independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $d[X_1;X_2] = k$\n* $I_1 := I_1 [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n* $I_2 := I[X_1+X_2 : X_1 + \\tilde X_1 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `second_estimate` : $$ I_2 \\leq 2 \\eta k + \\frac{2 \\eta (2 \\eta k - I_1)}{1 - \\eta}.$$\n-/\n\nopen MeasureTheory ProbabilityTheory ElementaryAddCommGroup\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2081', X\u2082'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- $$ d[X_1+\\tilde X_1; X_2+\\tilde X_2] \\geq k - \\frac{\\eta}{2} ( d[X_1; X_1] + d[X_2;X_2] ).$$\n-/\nlemma rdist_of_sums_ge' : d[X\u2081 + X\u2081' # X\u2082 + X\u2082'] \u2265 k - p.\u03b7 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082]) / 2 := by\n refine' LE.le.ge (LE.le.trans _ (distance_ge_of_min p h_min (hX\u2081.add hX\u2081') (hX\u2082.add hX\u2082')))\n rw [sub_sub, sub_le_sub_iff_left k, \u2190 mul_add,mul_div_assoc]\n refine' mul_le_mul_of_nonneg_left _ (by linarith [p.h\u03b7])\n have h\u2081' := condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2081' (h_indep.indepFun (show 0 \u2260 2 by decide))\n have h\u2082' := condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2082' (h_indep.indepFun (show 1 \u2260 3 by decide))\n rw [h\u2081.entropy_eq, add_sub_cancel_right, \u2190 (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081] at h\u2081'\n rw [h\u2082.entropy_eq, add_sub_cancel_right, \u2190 (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082] at h\u2082'\n linarith\n\n", "theoremStatement": "lemma second_estimate_aux :\n d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082] \u2264 2 * (k + (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7)) ", "theoremName": "second_estimate_aux", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "43796aefd15705ff6e03586a1ef09d82b3c5d56d", "date": "2023-12-14"}, "file": "PFR/PFR/SecondEstimate.lean", "module": "PFR.SecondEstimate", "jsonFile": "PFR.SecondEstimate.jsonl", "positionMetadata": {"lineInFile": 66, "tokenPositionInFile": 2827, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 16, "numPremises": 342, "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.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have hX\u2081_indep : IndepFun X\u2081 X\u2081' (\u03bc := \u2119) := h_indep.indepFun (show 0 \u2260 2 by decide)\n have hX\u2082_indep : IndepFun X\u2082 X\u2082' (\u03bc := \u2119) := h_indep.indepFun (show 1 \u2260 3 by decide)\n have hX_indep : IndepFun (X\u2081 + X\u2081') (X\u2082 + X\u2082') := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption) 0 2 1 3\n (by decide) (by decide) (by decide) (by decide)\n have h : d[X\u2081 + X\u2081' # X\u2082+ X\u2082'] \u2264 (2 + p.\u03b7) * k - (d[X\u2081# X\u2081] + d[X\u2082 # X\u2082]) / 2 - I\u2081 := by\n have h := hX_indep.rdist_eq (hX\u2081.add hX\u2081') (hX\u2082.add hX\u2082')\n rw [sub_eq_add (X\u2081 + X\u2081') (X\u2082 + X\u2082'), \u2190 sub_eq_add X\u2081 X\u2081', \u2190 sub_eq_add X\u2082 X\u2082',\n sub_eq_iff_eq_add.mp (sub_eq_iff_eq_add.mp (hX\u2081_indep.rdist_eq hX\u2081 hX\u2081').symm),\n sub_eq_iff_eq_add.mp (sub_eq_iff_eq_add.mp (hX\u2082_indep.rdist_eq hX\u2082 hX\u2082').symm),\n \u2190 h\u2081.entropy_eq, \u2190 h\u2082.entropy_eq, add_assoc, add_assoc, add_halves', add_halves',\n \u2190 (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081,\n \u2190 (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082,\n sub_eq_add X\u2081 X\u2081', sub_eq_add X\u2082 X\u2082', \u2190 add_assoc, add_right_comm _ X\u2081'] at h\n have h_indep' : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'] :=\n by exact h_indep.reindex_four_abdc\n have h' := ent_ofsum_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep' h_min\n convert (h.symm \u25b8 (sub_le_sub_right (sub_le_sub_right h' _) _)) using 1; ring\n have h' := (rdist_of_sums_ge' p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min).le.trans h\n rw [\u2190 div_le_iff' two_pos, \u2190 sub_le_iff_le_add', le_div_iff (by linarith [p.h\u03b7'])]\n linarith", "proofType": "tactic", "proofLengthLines": 21, "proofLengthTokens": 1550}} +{"srcContext": "import PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The fibring identity\n\nThe proof of the fibring identity, which is a key component of the proof of PFR.\n\n## Main statement\n\n* `sum_of_rdist_eq`: If $Y_1,Y_2,Y_3,Y_4$ are independent, then $d[Y_1; Y_2] + d[Y_3; Y_4]$ is equal to the sum of\n$$d[Y_1+Y_3; Y_2+Y_4] + d[Y_1|Y_1+Y_3; Y_2|Y_2+Y_4] $$\nand\n$$ I[Y_1+Y_2 : Y_2 + Y_4 | Y_1+Y_2+Y_3+Y_4].$$\n\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralFibring\n\n-- $\\pi : H \\to H'$ is a homomorphism between additive groups.\nvariable {H : Type*} [AddCommGroup H] [Countable H] [hH : MeasurableSpace H]\n [MeasurableSingletonClass H]\n {H' : Type*} [AddCommGroup H'] [Countable H'] [hH' : MeasurableSpace H']\n [MeasurableSingletonClass H']\n (\u03c0 : H \u2192+ H')\nvariable {\u03a9 \u03a9' : Type*} [m\u03a9 : MeasurableSpace \u03a9] [m\u03a9' : MeasurableSpace \u03a9']\n {\u03bc : Measure \u03a9} {\u03bc' : Measure \u03a9'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n", "theoremStatement": "/-- If $Z_1, Z_2$ are independent, then $d[Z_1; Z_2]$ is equal to\n$$ d[\\pi(Z_1);\\pi(Z_2)] + d[Z_1|\\pi(Z_1); Z_2 |\\pi(Z_2)]$$\nplus\n$$I( Z_1 - Z_2 : (\\pi(Z_1), \\pi(Z_2)) | \\pi(Z_1 - Z_2) ).$$\n-/\nlemma rdist_of_indep_eq_sum_fibre {Z_1 Z_2: \u03a9 \u2192 H} (h : IndepFun Z_1 Z_2 \u03bc)\n (h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2]:\n d[Z_1; \u03bc # Z_2; \u03bc] = d[\u03c0 \u2218 Z_1; \u03bc # \u03c0 \u2218 Z_2; \u03bc] + d[Z_1|\u03c0\u2218Z_1; \u03bc # Z_2|\u03c0\u2218Z_2; \u03bc] + I[Z_1-Z_2 : \u27e8\u03c0\u2218Z_1, \u03c0\u2218Z_2\u27e9 | \u03c0\u2218(Z_1 - Z_2); \u03bc] ", "theoremName": "rdist_of_indep_eq_sum_fibre", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "8269001e903a93e4eceb4336a8a20e1693618e0f", "date": "2023-11-18"}, "file": "PFR/PFR/Fibring.lean", "module": "PFR.Fibring", "jsonFile": "PFR.Fibring.jsonl", "positionMetadata": {"lineInFile": 30, "tokenPositionInFile": 916, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 15, "numPremises": 194, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", 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"Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", 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"Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h\u03c0 : Measurable \u03c0 := measurable_of_countable _\n have step1 : d[Z_1; \u03bc # Z_2; \u03bc] = d[\u03c0 \u2218 Z_1; \u03bc # \u03c0 \u2218 Z_2; \u03bc] +\n H[(Z_1 - Z_2)| \u03c0 \u2218 (Z_1 - Z_2); \u03bc] - H[Z_1 | \u03c0 \u2218 Z_1; \u03bc] / 2 - H[Z_2 | \u03c0 \u2218 Z_2; \u03bc] / 2 := by\n have hsub : H[(Z_1 - Z_2)| \u03c0 \u2218 (Z_1 - Z_2); \u03bc] = H[(Z_1 - Z_2); \u03bc] - H[\u03c0 \u2218 (Z_1 - Z_2); \u03bc] := condEntropy_comp_self (by measurability) h\u03c0\n\n rw [h.rdist_eq h1 h2, (h.comp h\u03c0 h\u03c0).rdist_eq (h\u03c0.comp h1) (h\u03c0.comp h2),\n condEntropy_comp_self h1 h\u03c0, condEntropy_comp_self h2 h\u03c0, hsub, map_comp_sub \u03c0]\n ring_nf\n have m0 : Measurable (fun x \u21a6 (x, \u03c0 x)) := measurable_of_countable _\n have h' : IndepFun (\u27e8Z_1, \u03c0 \u2218 Z_1\u27e9) (\u27e8Z_2, \u03c0 \u2218 Z_2\u27e9) \u03bc := h.comp m0 m0\n have m1 : Measurable (Z_1 - Z_2) := h1.sub h2\n have m2 : Measurable (\u27e8\u2191\u03c0 \u2218 Z_1, \u2191\u03c0 \u2218 Z_2\u27e9) := (h\u03c0.comp h1).prod_mk (h\u03c0.comp h2)\n have m3 : Measurable (\u2191\u03c0 \u2218 (Z_1 - Z_2)) := h\u03c0.comp m1\n have entroplem : H[Z_1 - Z_2|\u27e8\u27e8\u2191\u03c0 \u2218 Z_1, \u2191\u03c0 \u2218 Z_2\u27e9, \u2191\u03c0 \u2218 (Z_1 - Z_2)\u27e9; \u03bc]\n = H[Z_1 - Z_2|\u27e8\u2191\u03c0 \u2218 Z_1, \u2191\u03c0 \u2218 Z_2\u27e9; \u03bc] := by\n rw [map_comp_sub \u03c0]\n let f : H' \u00d7 H' \u2192 (H' \u00d7 H') \u00d7 H' := fun (x,y) \u21a6 ((x,y), x - y)\n have hf : Function.Injective f := fun _ _ h \u21a6 (Prod.ext_iff.1 h).1\n have mf : Measurable f := measurable_id.prod_mk measurable_sub\n refine condEntropy_of_injective' \u03bc m1 m2 f hf (mf.comp m2)\n rw [step1, condMutualInfo_eq' m1 m2 m3, entroplem,\n condRuzsaDist_of_indep h1 (h\u03c0.comp h1) h2 (h\u03c0.comp h2) \u03bc h']\n ring_nf", "proofType": "tactic", "proofLengthLines": 23, "proofLengthTokens": 1419}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\nopen ElementaryAddCommGroup\n\n/-- In fact $k$ is at most\n $$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\nlemma construct_good_improved' :\n k \u2264 \u03b4 + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 # T\u2081 | T\u2082] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2082 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2082 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2083 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2083 | T\u2082] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2081 | T\u2082] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2081 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2083 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2083 | T\u2082] - d[p.X\u2080\u2082 # X\u2082])) := by\n have I1 : I[T\u2082 : T\u2081] = I[T\u2081 : T\u2082] := mutualInfo_comm hT\u2082 hT\u2081 _\n have I2 : I[T\u2083 : T\u2081] = I[T\u2081 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2081 _\n have I3 : I[T\u2083 : T\u2082] = I[T\u2082 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2082 _\n have Z123 := construct_good_prelim' h_min hT hT\u2081 hT\u2082 hT\u2083\n have h132 : T\u2081 + T\u2083 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z132 := construct_good_prelim' h_min h132 hT\u2081 hT\u2083 hT\u2082\n have h213 : T\u2082 + T\u2081 + T\u2083 = 0 := by rw [\u2190 hT]; abel\n have Z213 := construct_good_prelim' h_min h213 hT\u2082 hT\u2081 hT\u2083\n have h231 : T\u2082 + T\u2083 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z231 := construct_good_prelim' h_min h231 hT\u2082 hT\u2083 hT\u2081\n have h312 : T\u2083 + T\u2081 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z312 := construct_good_prelim' h_min h312 hT\u2083 hT\u2081 hT\u2082\n have h321 : T\u2083 + T\u2082 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z321 := construct_good_prelim' h_min h321 hT\u2083 hT\u2082 hT\u2081\n simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321\n linarith\n\n/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\nlemma construct_good_improved'' {\u03a9' : Type*} [MeasurableSpace \u03a9'] (\u03bc : Measure \u03a9')\n [IsProbabilityMeasure \u03bc] {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G}\n (hT : T\u2081 + T\u2082 + T\u2083 = 0) (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083) :\n k \u2264 I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc] + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]))\n := by\n let M : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n exact construct_good_improved' h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend aux\n\nopen scoped BigOperators\n\n", "theoremStatement": "/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\nlemma averaged_construct_good : k \u2264 (I[U : V | S] + I[V : W | S] + I[W : U | S])\n + (p.\u03b7 / 6) * (((d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]))\n + ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])))\n ", "theoremName": "averaged_construct_good", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 421, "tokenPositionInFile": 21597, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, 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"Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "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.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.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.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.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.Paths", "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", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "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", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "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", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "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", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have hS : Measurable S := ((hX\u2081.add' hX\u2082).add' hX\u2081').add' hX\u2082'\n have hU : Measurable U := hX\u2081.add' hX\u2082\n have hV : Measurable V := hX\u2081'.add hX\u2082\n have hW : Measurable W := hX\u2081'.add hX\u2081\n have hUVW : U + V + W = 0 := sum_uvw_eq_zero X\u2081 X\u2082 X\u2081'\n have hz (a : \u211d) : a = \u2211 z, (\u2119 (S \u207b\u00b9' {z})).toReal * a := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u2119 hS, one_mul]\n rw [hz k, hz (d[p.X\u2080\u2081 # X\u2081]), hz (d[p.X\u2080\u2082 # X\u2082])]\n simp only [condMutualInfo_eq_sum' hS, \u2190 Finset.sum_add_distrib, \u2190 mul_add,\n condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, \u2190 Finset.sum_sub_distrib, \u2190 mul_sub, Finset.mul_sum,\n \u2190 mul_assoc (p.\u03b7/6), mul_comm (p.\u03b7/6), mul_assoc _ _ (p.\u03b7/6)]\n rw [Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_le_sum (fun i _hi \u21a6 ?_)\n rcases eq_or_ne (\u2119 (S \u207b\u00b9' {i})) 0 with h'i|h'i\n \u00b7 simp [h'i]\n rw [mul_assoc, \u2190 mul_add]\n gcongr\n have : IsProbabilityMeasure (\u2119[|S \u207b\u00b9' {i}]) := cond_isProbabilityMeasure \u2119 h'i\n linarith [construct_good_improved'' h_min (\u2119[|S \u207b\u00b9' {i}]) hUVW hU hV hW]", "proofType": "tactic", "proofLengthLines": 19, "proofLengthTokens": 1030}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\nopen ElementaryAddCommGroup\n\n/-- In fact $k$ is at most\n $$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\nlemma construct_good_improved' :\n k \u2264 \u03b4 + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 # T\u2081 | T\u2082] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2082 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2082 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2083 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2083 | T\u2082] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2081 | T\u2082] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2081 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2083 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2083 | T\u2082] - d[p.X\u2080\u2082 # X\u2082])) := by\n have I1 : I[T\u2082 : T\u2081] = I[T\u2081 : T\u2082] := mutualInfo_comm hT\u2082 hT\u2081 _\n have I2 : I[T\u2083 : T\u2081] = I[T\u2081 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2081 _\n have I3 : I[T\u2083 : T\u2082] = I[T\u2082 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2082 _\n have Z123 := construct_good_prelim' h_min hT hT\u2081 hT\u2082 hT\u2083\n have h132 : T\u2081 + T\u2083 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z132 := construct_good_prelim' h_min h132 hT\u2081 hT\u2083 hT\u2082\n have h213 : T\u2082 + T\u2081 + T\u2083 = 0 := by rw [\u2190 hT]; abel\n have Z213 := construct_good_prelim' h_min h213 hT\u2082 hT\u2081 hT\u2083\n have h231 : T\u2082 + T\u2083 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z231 := construct_good_prelim' h_min h231 hT\u2082 hT\u2083 hT\u2081\n have h312 : T\u2083 + T\u2081 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z312 := construct_good_prelim' h_min h312 hT\u2083 hT\u2081 hT\u2082\n have h321 : T\u2083 + T\u2082 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z321 := construct_good_prelim' h_min h321 hT\u2083 hT\u2082 hT\u2081\n simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321\n linarith\n\n/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\nlemma construct_good_improved'' {\u03a9' : Type*} [MeasurableSpace \u03a9'] (\u03bc : Measure \u03a9')\n [IsProbabilityMeasure \u03bc] {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G}\n (hT : T\u2081 + T\u2082 + T\u2083 = 0) (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083) :\n k \u2264 I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc] + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]))\n := by\n let M : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n exact construct_good_improved' h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend aux\n\nopen scoped BigOperators\n\n/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\nlemma averaged_construct_good : k \u2264 (I[U : V | S] + I[V : W | S] + I[W : U | S])\n + (p.\u03b7 / 6) * (((d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]))\n + ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])))\n := by\n have hS : Measurable S := ((hX\u2081.add' hX\u2082).add' hX\u2081').add' hX\u2082'\n have hU : Measurable U := hX\u2081.add' hX\u2082\n have hV : Measurable V := hX\u2081'.add hX\u2082\n have hW : Measurable W := hX\u2081'.add hX\u2081\n have hUVW : U + V + W = 0 := sum_uvw_eq_zero X\u2081 X\u2082 X\u2081'\n have hz (a : \u211d) : a = \u2211 z, (\u2119 (S \u207b\u00b9' {z})).toReal * a := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u2119 hS, one_mul]\n rw [hz k, hz (d[p.X\u2080\u2081 # X\u2081]), hz (d[p.X\u2080\u2082 # X\u2082])]\n simp only [condMutualInfo_eq_sum' hS, \u2190 Finset.sum_add_distrib, \u2190 mul_add,\n condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, \u2190 Finset.sum_sub_distrib, \u2190 mul_sub, Finset.mul_sum,\n \u2190 mul_assoc (p.\u03b7/6), mul_comm (p.\u03b7/6), mul_assoc _ _ (p.\u03b7/6)]\n rw [Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_le_sum (fun i _hi \u21a6 ?_)\n rcases eq_or_ne (\u2119 (S \u207b\u00b9' {i})) 0 with h'i|h'i\n \u00b7 simp [h'i]\n rw [mul_assoc, \u2190 mul_add]\n gcongr\n have : IsProbabilityMeasure (\u2119[|S \u207b\u00b9' {i}]) := cond_isProbabilityMeasure \u2119 h'i\n linarith [construct_good_improved'' h_min (\u2119[|S \u207b\u00b9' {i}]) hUVW hU hV hW]\n\nvariable (p)\n\nlemma dist_diff_bound_1 :\n (d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n \u2264 (16 * k + 6 * d[X\u2081 # X\u2081] + 2 * d[X\u2082 # X\u2082]) / 4 + (H[X\u2081 + X\u2081'] - H[X\u2082 + X\u2082']) / 4\n + (H[X\u2082 | X\u2082 + X\u2082'] - H[X\u2081 | X\u2081 + X\u2081']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2082' X\u2081' hX\u2081 hX\u2082 hX\u2082' hX\u2081' h_indep.reindex_four_abcd\n have I2 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep.reindex_four_abdc\n have I3 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2082 X\u2081' hX\u2081 hX\u2082' hX\u2082 hX\u2081' h_indep.reindex_four_acbd\n have I4 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2081' X\u2082 hX\u2081 hX\u2082' hX\u2081' hX\u2082 h_indep.reindex_four_acdb\n have I5 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082 X\u2082' hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_adbc\n have I6 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082' X\u2082 hX\u2081 hX\u2081' hX\u2082' hX\u2082 h_indep.reindex_four_adcb\n have C1 : U + X\u2082' + X\u2081' = S := by abel\n have C2 : W + X\u2082 + X\u2082' = S := by abel\n have C3 : X\u2081 + X\u2082' + X\u2082 + X\u2081' = S := by abel\n have C4 : X\u2081 + X\u2082' + X\u2081' + X\u2082 = S := by abel\n have C5 : W + X\u2082' + X\u2082 = S := by abel\n have C7 : X\u2082 + X\u2081' = V := by abel\n have C8 : X\u2081 + X\u2081' = W := by abel\n have C9 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082.symm\n have C10 : d[X\u2082 # X\u2081'] = d[X\u2081' # X\u2082] := rdist_symm\n have C11 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081.symm\n have C12 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq h\u2082.symm\n have C13 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] := (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082.symm\n have C14 : d[X\u2081' # X\u2082] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C15 : H[X\u2081' + X\u2082'] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C16 : H[X\u2082'] = H[X\u2082] := h\u2082.symm.entropy_eq\n have C17 : H[X\u2081'] = H[X\u2081] := h\u2081.symm.entropy_eq\n have C18 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C19 : H[X\u2082' + X\u2081'] = H[U] := by rw [add_comm]; exact C15\n have C20 : d[X\u2082' # X\u2082] = d[X\u2082 # X\u2082] := h\u2082.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C21 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C22 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C23 : X\u2082' + X\u2082 = X\u2082 + X\u2082' := by abel\n have C24 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C25 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C27 : H[X\u2082' | X\u2082 + X\u2082'] = H[X\u2082 | X\u2082 + X\u2082'] := by\n conv_lhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2082'.add' hX\u2082) hX\u2082 (hX\u2082.add' hX\u2082')\n have I : IdentDistrib (\u27e8X\u2082', X\u2082\u27e9) (\u27e8X\u2082, X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2082.symm h\u2082\n (h_indep.indepFun (show 2 \u2260 1 by decide)) (h_indep.indepFun (show 1 \u2260 2 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C28 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C29 : H[X\u2081' | V] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082) hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C30 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n simp only [C1, C2, C3, C4, C5, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30] at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma dist_diff_bound_2 :\n ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]))\n \u2264 (16 * k + 6 * d[X\u2082 # X\u2082] + 2 * d[X\u2081 # X\u2081]) / 4 + (H[X\u2082 + X\u2082'] - H[X\u2081 + X\u2081']) / 4\n + (H[X\u2081 | X\u2081 + X\u2081'] - H[X\u2082 | X\u2082 + X\u2082']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2082' X\u2081' hX\u2082 hX\u2081 hX\u2082' hX\u2081' h_indep.reindex_four_bacd\n have I2 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2081' X\u2082' hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_badc\n have I3 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081 X\u2081' hX\u2082 hX\u2082' hX\u2081 hX\u2081' h_indep.reindex_four_bcad\n have I4 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081' X\u2081 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bcda\n have I5 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2081 X\u2082' hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bdac\n have I6 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2082' X\u2081 hX\u2082 hX\u2081' hX\u2082' hX\u2081 h_indep.reindex_four_bdca\n have C1 : X\u2082 + X\u2081 = X\u2081 + X\u2082 := by abel\n have C2 : X\u2081 + X\u2081' = W := by abel\n have C3 : U + X\u2082' + X\u2081' = S := by abel\n have C4 : X\u2082 + X\u2081' = V := by abel\n have C5 : X\u2082 + X\u2082' + X\u2081 + X\u2081' = S := by abel\n have C6 : X\u2082 + X\u2082' + X\u2081' + X\u2081 = S := by abel\n have C7 : V + X\u2081 + X\u2082' = S := by abel\n have C8 : V + X\u2082' + X\u2081 = S := by abel\n have C9 : d[X\u2082 # X\u2081] = d[X\u2081 # X\u2082] := rdist_symm\n have C10 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n have C11 : d[X\u2082 # X\u2081'] = d[X\u2081 # X\u2082] := by\n rw [rdist_symm]\n exact ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n have C12 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C13 : d[X\u2082' # X\u2081] = d[X\u2081 # X\u2082'] := rdist_symm\n have C14 : d[X\u2081' # X\u2081] = d[X\u2081 # X\u2081'] := rdist_symm\n have C15 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n have C16 : H[X\u2081' + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C17 : H[X\u2082' + X\u2081'] = H[X\u2081 + X\u2082] := by rw [add_comm]; exact C16\n have C18 : H[X\u2081'] = H[X\u2081] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2081.symm\n have C19 : H[X\u2082'] = H[X\u2082] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2082.symm\n have C20 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C21 : H[X\u2081' | W] = H[X\u2081 | W] := by\n conv_rhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2081) hX\u2081 (hX\u2081.add' hX\u2081')\n have I : IdentDistrib (\u27e8X\u2081', X\u2081\u27e9) (\u27e8X\u2081, X\u2081'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2081\n (h_indep.indepFun (show 3 \u2260 0 by decide)) (h_indep.indepFun (show 0 \u2260 3 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C22 : H[X\u2082' | X\u2082' + X\u2081] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C23 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C24 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C25 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C27 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n have C28 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C29 : H[X\u2082' + X\u2081] = H[X\u2081 + X\u2082] := by\n rw [add_comm]\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C30 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2081.symm\n have C31 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable) h\u2082.symm\n simp only [C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C25, C26, C27, C28, C29, C30, C31]\n at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\n", "theoremStatement": "lemma averaged_final : k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) ", "theoremName": "averaged_final", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "43796aefd15705ff6e03586a1ef09d82b3c5d56d", "date": "2023-12-14"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 660, "tokenPositionInFile": 36075, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 18, "numPremises": 240, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", 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"Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "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.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.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.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.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.Paths", "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", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "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", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "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", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "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", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply (averaged_construct_good hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_min).trans\n have : 0 \u2264 p.\u03b7 := p.h\u03b7.le\n have := sum_condMutual_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep.reindex_four_abdc h_min\n gcongr ?_ + (p.\u03b7 / 6) * ?_\n linarith [dist_diff_bound_1 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep,\n dist_diff_bound_2 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep]", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 348}} +{"srcContext": "import PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The 100% version of entropic PFR\n\nHere we show entropic PFR in the case of doubling constant zero.\n\n## Main results\n\n* `exists_isUniform_of_rdist_eq_zero` : If $d[X_1;X_2]=0$, then there exists a subgroup $H \\leq G$ such that $d[X_1;U_H] = d[X_2;U_H] = 0$.\n-/\n\nopen MeasureTheory ProbabilityTheory Real\n\nvariable {\u03a9 : Type*} {G : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd\u2082 G] [MeasurableSub\u2082 G] {X : \u03a9 \u2192 G}\n\n/-- The symmetry group Sym of $X$: the set of all $h \u2208 G$ such that $X + h$ has an identical\ndistribution to $X$. -/\ndef symmGroup (X : \u03a9 \u2192 G) (hX : Measurable X) : AddSubgroup G where\n carrier := {x | IdentDistrib X (fun \u03c9 \u21a6 X \u03c9 + x)}\n add_mem' := by\n intro x y hx hy\n let f : G \u2192 G := fun g \u21a6 g + x\n have : IdentDistrib (f \u2218 X) (fun \u03c9 \u21a6 f (X \u03c9 + y)) := hy.comp $ measurable_add_const _\n have Z := hx.trans this\n dsimp\n convert Z using 1\n ext \u03c9\n simp only [f]\n abel\n zero_mem' := by simpa using IdentDistrib.refl hX.aemeasurable\n neg_mem' := by\n intro x hx\n dsimp at hx \u22a2\n let f : G \u2192 G := fun g \u21a6 g - x\n have : IdentDistrib (f \u2218 X) (fun \u03c9 \u21a6 f (X \u03c9 + x)) := hx.comp $ measurable_sub_const' _\n convert this.symm using 1\n \u00b7 ext \u03c9\n simp only [f]\n abel\n \u00b7 ext \u03c9\n simp only [Function.comp_apply]\n abel_nf\n\n@[simp] lemma mem_symmGroup (hX : Measurable X) {x : G} :\n x \u2208 symmGroup X hX \u2194 IdentDistrib X (fun \u03c9 \u21a6 X \u03c9 + x) := Iff.rfl\n\nlemma ProbabilityTheory.IdentDistrib.symmGroup_eq {\u03a9' : Type*} [MeasureSpace \u03a9'] {X' : \u03a9' \u2192 G}\n (hX : Measurable X) (hX' : Measurable X') (h : IdentDistrib X X') :\n symmGroup X hX = symmGroup X' hX' := by\n ext x\n have A : Measurable (fun a \u21a6 a + x) := measurable_add_const _\n exact \u27e8fun H \u21a6 h.symm.trans (H.trans (h.comp A)), fun H \u21a6 h.trans (H.trans (h.symm.comp A))\u27e9\n\nvariable [MeasurableSingletonClass G]\n\n", "theoremStatement": "/-- If $d[X ;X]=0$, and $x,y \\in G$ are such that $P[X=x], P[X=y]>0$,\nthen $x-y \\in \\mathrm{Sym}[X]$. -/\nlemma sub_mem_symmGroup (hX : Measurable X) (hdist : d[X # X] = 0)\n {x y : G} (hx : \u2119 (X\u207b\u00b9' {x}) \u2260 0) (hy : \u2119 (X\u207b\u00b9' {y}) \u2260 0) : x - y \u2208 symmGroup X hX ", "theoremName": "sub_mem_symmGroup", "fileCreated": {"commit": "4a43f679d5bb73bbd9653c2f286f5b389218760c", "date": "2023-11-16"}, "theoremCreated": {"commit": "758fe36f15c65ff90cff7cf8e7722fd5c38ede74", "date": "2023-11-22"}, "file": "PFR/PFR/HundredPercent.lean", "module": "PFR.HundredPercent", "jsonFile": "PFR.HundredPercent.jsonl", "positionMetadata": {"lineInFile": 58, "tokenPositionInFile": 1962, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 16, "numPremises": 319, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", 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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.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", 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"Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", 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"Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", 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"Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n /- Consider two independent copies `X'` and `Y'` of `X`. The assumption on the Rusza distance\n ensures that `H[X' - Y' | Y'] = H[X' - Y']`, i.e., `X' - Y'` and `Y'` are independent. Therefore,\n the distribution of `X' - c` is independent of `c` for `c` in the support of `Y'`.\n In particular, `X' - x` and `X' - y` have the same distribution, which is equivalent to the\n claim of the lemma. -/\n rcases ProbabilityTheory.independent_copies_two hX hX with\n \u27e8\u03a9', m\u03a9', X', Y', hP, hX', hY', hindep, hidX, hidY\u27e9\n rw [hidX.symm.symmGroup_eq hX hX']\n have A : H[X' - Y' | Y'] = H[X' - Y'] := calc\n H[X' - Y' | Y'] = H[X' | Y'] := condEntropy_sub_right hX' hY'\n _ = H[X'] := hindep.condEntropy_eq_entropy hX' hY'\n _ = H[X' - Y'] := by\n have : d[X' # Y'] = 0 := by rwa [hidX.rdist_eq hidY]\n rw [hindep.rdist_eq hX' hY', \u2190 (hidX.trans hidY.symm).entropy_eq] at this\n linarith\n have I : IndepFun (X' - Y') Y' := by\n refine (mutualInfo_eq_zero (hX'.sub' hY') hY').1 ?_\n rw [mutualInfo_eq_entropy_sub_condEntropy (hX'.sub' hY') hY', A, sub_self]\n have M : \u2200 c, \u2119 (Y' \u207b\u00b9' {c}) \u2260 0 \u2192 IdentDistrib (fun \u03c9 \u21a6 X' \u03c9 - c) (X' - Y') := by\n intro c hc\n let F := fun \u03c9 \u21a6 X' \u03c9 - c\n refine \u27e8(hX'.sub_const _).aemeasurable, (hX'.sub' hY').aemeasurable, ?_\u27e9\n ext s hs\n rw [Measure.map_apply (hX'.sub_const _) hs, Measure.map_apply (hX'.sub' hY') hs]\n have : \u2119 (F \u207b\u00b9' s) * \u2119 (Y' \u207b\u00b9' {c}) = \u2119 ((X' - Y') \u207b\u00b9' s) * \u2119 (Y' \u207b\u00b9' {c}) := by calc\n \u2119 (F \u207b\u00b9' s) * \u2119 (Y' \u207b\u00b9' {c}) = \u2119 (F \u207b\u00b9' s \u2229 Y' \u207b\u00b9' {c}) := by\n have hFY' : IndepFun F Y' := by\n have : Measurable (fun z \u21a6 z - c) := measurable_sub_const' c\n apply hindep.comp this measurable_id\n rw [indepFun_iff_measure_inter_preimage_eq_mul.1 hFY' _ _ hs $ measurableSet_discrete _]\n _ = \u2119 ((X' - Y') \u207b\u00b9' s \u2229 Y' \u207b\u00b9' {c}) := by\n congr 1; ext z; simp (config := {contextual := true})\n _ = \u2119 ((X' - Y') \u207b\u00b9' s) * \u2119 (Y' \u207b\u00b9' {c}) := by\n rw [indepFun_iff_measure_inter_preimage_eq_mul.1 I _ _ hs $ measurableSet_discrete _]\n rwa [ENNReal.mul_eq_mul_right hc (measure_ne_top \u2119 _)] at this\n have J : IdentDistrib (fun \u03c9 \u21a6 X' \u03c9 - x) (fun \u03c9 \u21a6 X' \u03c9 - y) := by\n have Px : \u2119 (Y' \u207b\u00b9' {x}) \u2260 0 := by\n convert hx; exact hidY.measure_mem_eq $ measurableSet_discrete _\n have Py : \u2119 (Y' \u207b\u00b9' {y}) \u2260 0 := by\n convert hy; exact hidY.measure_mem_eq $ measurableSet_discrete _\n exact (M x Px).trans (M y Py).symm\n have : IdentDistrib X' (fun \u03c9 \u21a6 X' \u03c9 + (x - y)) := by\n have : Measurable (fun c \u21a6 c + x) := measurable_add_const x\n convert J.comp this using 1\n \u00b7 ext \u03c9; simp\n \u00b7 ext \u03c9; simp; abel\n exact this", "proofType": "tactic", "proofLengthLines": 47, "proofLengthTokens": 2658}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\nlemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal := by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]\n\nlemma entropy_eq_sum_finset' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finset hX hA\n\nlemma entropy_eq_sum_finiteRange (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog (\u03bc.map X {x}).toReal := entropy_eq_sum_finset hX (A := FiniteRange.toFinset X) (full_measure_of_finiteRange hX)\n\nlemma entropy_eq_sum_finiteRange' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finiteRange hX\n\n/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\nlemma entropy_cond_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) :\n H[X | Y \u2190 y ; \u03bc] = \u2211' x, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum hX]\n\nlemma entropy_cond_eq_sum_finiteRange (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) [FiniteRange X]:\n H[X | Y \u2190 y ; \u03bc] = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum_finiteRange hX]\n\n/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\nlemma entropy_comp_of_injective\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 T) (hf : Function.Injective f) :\n H[f \u2218 X ; \u03bc] = H[X ; \u03bc] := by\n have hf_m : Measurable f := measurable_of_countable f\n rw [entropy_def, \u2190 Measure.map_map hf_m hX, measureEntropy_map_of_injective _ _ hf_m hf,\n entropy_def]\n\n\n/-- The entropy of any constant is zero. -/\n@[simp] lemma entropy_const [IsProbabilityMeasure \u03bc] (c : S) : H[fun _ \u21a6 c ; \u03bc] = 0 := by\n simp [entropy, Measure.map_const]\n\nopen Set\n\nopen Function\n\n/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\nlemma IsUniform.entropy_eq (H : Finset S) (X : \u03a9 \u2192 S) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n haveI : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX'.aemeasurable\n have (t : S) : negMulLog ((\u03bc.map X).real {t}) = ((\u03bc.map X).real {t}) * log (Nat.card H) := by\n by_cases ht : t \u2208 H\n \u00b7 simp only [negMulLog, neg_mul, neg_mul_eq_mul_neg, IsUniform.measureReal_preimage_of_mem'\n hX hX' ht, one_div, log_inv, neg_neg]\n \u00b7 simp only [negMulLog, map_measureReal_apply hX' (MeasurableSet.singleton t),\n IsUniform.measureReal_preimage_of_nmem hX ht, neg_zero, log_zero, mul_zero, zero_mul]\n rw [entropy_eq_sum_finset' (A := H) hX', Finset.sum_congr rfl (fun t _ \u21a6 this t), \u2190 Finset.sum_mul,\n Finset.sum_realMeasure_singleton]\n . convert one_mul _\n have := IsUniform.full_measure hX hX'\n simp at this\n rw [Measure.real, this]\n simp\n rw [Measure.map_apply hX' (by measurability)]\n exact hX.measure_preimage_compl\n\n\n/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\nlemma IsUniform.entropy_eq' {H : Set S} [Finite H] {X : \u03a9 \u2192 S} {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n convert IsUniform.entropy_eq H.toFinite.toFinset X ?_ hX' with x\n . simp; exact Iff.rfl\n . infer_instance\n convert hX\n simp\n\n/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\nlemma entropy_eq_log_card {X : \u03a9 \u2192 S} [Fintype S] (hX : Measurable X) (\u03bc : Measure \u03a9) [h\u03bc : NeZero \u03bc]\n [IsFiniteMeasure \u03bc] :\n entropy X \u03bc = log (Fintype.card S) \u2194 \u2200 s, \u03bc.map X {s} = \u03bc Set.univ / Fintype.card S := by\n rcases eq_zero_or_neZero (\u03bc.map X) with h | h\n . have := Measure.le_map_apply (@Measurable.aemeasurable \u03a9 S _ _ X \u03bc hX) Set.univ\n simp [h] at this; simp [this] at h\u03bc\n have : IsFiniteMeasure (\u03bc.map X) := by\n apply Measure.isFiniteMeasure_map\n rw [entropy_def, measureEntropy_eq_card_iff_measure_eq, Measure.map_apply hX MeasurableSet.univ]\n simp\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\nlemma prob_ge_exp_neg_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) [hX': FiniteRange X] :\n \u2203 s : S, \u03bc.map X {s} \u2265 (\u03bc Set.univ) * (rexp (- H[X ; \u03bc])).toNNReal := by\n let \u03bcS := \u03bc.map X\n let \u03bcs s := \u03bcS {s}\n rcases finiteSupport_of_finiteRange (X := X) with \u27e8A, hA\u27e9\n\n let S_nonzero := A.filter (fun s \u21a6 \u03bcs s \u2260 0)\n\n set norm := \u03bcS A with rw_norm\n have h_norm : norm = \u03bc Set.univ := by\n have := measure_add_measure_compl (\u03bc := \u03bcS) (s := A) (Finset.measurableSet _)\n rw [hA, add_zero] at this\n simp [norm, \u03bcS, this, Measure.map_apply hX MeasurableSet.univ]\n\n let pdf_nn s := norm\u207b\u00b9 * \u03bcs s\n let pdf s := (pdf_nn s).toReal\n let neg_log_pdf s := -log (pdf s)\n\n rcases Finset.eq_empty_or_nonempty S_nonzero with h_empty | h_nonempty\n . have h_norm_zero : \u03bc Set.univ = 0 := by\n have h : \u2200 s \u2208 A, \u03bcs s \u2260 0 \u2192 \u03bcs s \u2260 0 := fun _ _ h \u21a6 h\n rw [\u2190 h_norm, rw_norm, \u2190 Finset.sum_measure_singleton, \u2190 Finset.sum_filter_of_ne h,\n show Finset.filter _ _ = S_nonzero from rfl, h_empty, show Finset.sum \u2205 \u03bcs = 0 from rfl]\n use Classical.arbitrary (\u03b1 := S)\n rw [h_norm_zero, zero_mul]\n exact le_of_not_gt ENNReal.not_lt_zero\n\n rcases exists_or_forall_not (fun s \u21a6 \u03bc.map X {s} = \u221e) with h_infty | h_finite\n . obtain \u27e8s, h_s\u27e9 := h_infty\n use s; rw [h_s] ; exact le_top\n\n rcases eq_zero_or_neZero \u03bc with h_zero_measure | _\n . use Classical.arbitrary (\u03b1 := S)\n rw [h_zero_measure, show (0 : Measure \u03a9) _ = 0 from rfl, zero_mul]\n exact zero_le _\n\n have h_norm_pos : 0 < norm := by\n rw [h_norm, Measure.measure_univ_pos]\n exact NeZero.ne \u03bc\n have h_norm_finite : norm < \u221e := by\n rw [rw_norm, \u2190 Finset.sum_measure_singleton]\n exact ENNReal.sum_lt_top (fun s _ \u21a6 h_finite s)\n have h_invinvnorm_finite : norm\u207b\u00b9\u207b\u00b9 \u2260 \u221e := by\n rw [inv_inv]\n exact LT.lt.ne_top h_norm_finite\n have h_invnorm_ne_zero : norm\u207b\u00b9 \u2260 0 := ENNReal.inv_ne_top.mp h_invinvnorm_finite\n have h_invnorm_finite : norm\u207b\u00b9 \u2260 \u221e := by\n rw [\u2190 ENNReal.inv_ne_zero, inv_inv]\n exact ne_zero_of_lt h_norm_pos\n have h_pdf_finite : \u2200 s, pdf_nn s \u2260 \u221e := fun s \u21a6 ENNReal.mul_ne_top h_invnorm_finite (h_finite s)\n\n have h_norm_cancel : norm * norm\u207b\u00b9 = 1 :=\n ENNReal.mul_inv_cancel (ne_zero_of_lt h_norm_pos) (LT.lt.ne_top h_norm_finite)\n have h_pdf1 : (\u2211 s in A, pdf s) = 1 := by\n rw [\u2190 ENNReal.toReal_sum (fun s _ \u21a6 h_pdf_finite s), \u2190 Finset.mul_sum,\n Finset.sum_measure_singleton, mul_comm, h_norm_cancel, ENNReal.one_toReal]\n\n let \u27e8s_max, hs, h_min\u27e9 := Finset.exists_min_image S_nonzero neg_log_pdf h_nonempty\n have h_pdf_s_max_pos : 0 < pdf s_max := by\n rw [Finset.mem_filter] at hs\n have h_nonzero : pdf s_max \u2260 0 := ENNReal.toReal_ne_zero.mpr\n \u27e8mul_ne_zero h_invnorm_ne_zero hs.2, ENNReal.mul_ne_top h_invnorm_finite (h_finite s_max)\u27e9\n exact LE.le.lt_of_ne ENNReal.toReal_nonneg h_nonzero.symm\n\n use s_max\n rw [\u2190 h_norm, \u2190 one_mul (\u03bc.map X _), \u2190 h_norm_cancel, mul_assoc]\n apply mul_le_mul_of_nonneg_left _ (le_of_lt h_norm_pos)\n show ENNReal.ofReal (rexp (-H[X ; \u03bc])) \u2264 pdf_nn s_max\n rw [ENNReal.ofReal_le_iff_le_toReal (h_pdf_finite _),\n show (pdf_nn _).toReal = pdf _ from rfl, \u2190 Real.exp_log h_pdf_s_max_pos]\n apply exp_monotone\n rw [neg_le, \u2190 one_mul (-log _), \u2190 h_pdf1, Finset.sum_mul]\n let g_lhs s := pdf s * neg_log_pdf s_max\n let g_rhs s := -pdf s * log (pdf s)\n suffices \u2211 s in A, g_lhs s \u2264 \u2211 s in A, g_rhs s by\n convert this\n rw [entropy_eq_sum_finset hX hA]\n congr with s\n simp only [negMulLog, neg_mul, ENNReal.toReal_mul, neg_inj, g_rhs, pdf, pdf_nn]\n simp at h_norm\n rw [h_norm]\n simp\n have h_lhs : \u2200 s, \u03bcs s = 0 \u2192 g_lhs s = 0 := by {intros _ h; simp [g_lhs, pdf, pdf_nn, h]}\n have h_rhs : \u2200 s, \u03bcs s = 0 \u2192 g_rhs s = 0 := by {intros _ h; simp [g_rhs, pdf, pdf_nn, h]}\n rw [\u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_lhs s).mt),\n \u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_rhs s).mt)]\n apply Finset.sum_le_sum\n intros s h_s\n rw [show g_lhs s = _ * _ from rfl, show g_rhs s = _ * _ from rfl, neg_mul_comm]\n exact mul_le_mul_of_nonneg_left (h_min s h_s) ENNReal.toReal_nonneg\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\nlemma prob_ge_exp_neg_entropy' {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (X : \u03a9 \u2192 S) (hX : Measurable X) [FiniteRange X] :\n \u2203 s : S, rexp (- H[X ; \u03bc]) \u2264 \u03bc.real (X \u207b\u00b9' {s}) := by\n obtain \u27e8s, hs\u27e9 := prob_ge_exp_neg_entropy X \u03bc hX\n use s\n rwa [IsProbabilityMeasure.measure_univ, one_mul, ge_iff_le,\n (show ENNReal.ofNNReal _ = ENNReal.ofReal _ from rfl),\n ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _), \u2190 Measure.real,\n map_measureReal_apply hX (MeasurableSet.singleton s)] at hs\n\n/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\nlemma const_of_nonpos_entropy {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] {X : \u03a9 \u2192 S} (hX : Measurable X) [FiniteRange X] (hent: H[X; \u03bc] \u2264 0):\n \u2203 s : S, \u03bc.real (X \u207b\u00b9' {s}) = 1 := by\n rcases prob_ge_exp_neg_entropy' (\u03bc := \u03bc) X hX with \u27e8 s, hs \u27e9\n use s\n apply LE.le.antisymm\n . rw [\u2190 IsProbabilityMeasure.measureReal_univ (\u03bc := \u03bc)]\n exact measureReal_mono (subset_univ _) (by finiteness)\n refine le_trans ?_ hs\n simp [hent]\n\n\n/-- $H[X, Y] = H[Y, X]$. -/\nlemma entropy_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n change H[Prod.swap \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hY.prod_mk hX) Prod.swap Prod.swap_injective\n\n/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\nlemma entropy_assoc (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n change H[MeasurableEquiv.prodAssoc \u2218 \u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc ((hX.prod_mk hY).prod_mk hZ) _ $ Equiv.injective _\n\n/-- $H[X, f(X)] = H[X]$.-/\n@[simp] lemma entropy_prod_comp (hX : Measurable X) (\u03bc : Measure \u03a9) (f : S \u2192 T) :\n H[\u27e8X, f \u2218 X\u27e9; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) fun _ _ ab \u21a6 (Prod.ext_iff.1 ab).1\n\nend entropy\n\nsection condEntropy\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n\n/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\nnoncomputable\ndef condEntropy (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]]\n\nlemma condEntropy_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n condEntropy X Y \u03bc = (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]] := rfl\n\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \" ; \" \u03bc \"]\" => condEntropy X Y \u03bc\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \"]\" => condEntropy X Y volume\n\nlemma condEntropy_eq_zero\n (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (t : T) (ht : (\u03bc.map Y {t}).toReal = 0) : H[X | Y \u2190 t ; \u03bc] = 0 := by\n convert entropy_zero_measure X\n apply cond_eq_zero_of_meas_eq_zero\n rw [Measure.map_apply hY (MeasurableSet.singleton t)] at ht\n rw [\u2190 measureReal_eq_zero_iff]\n exact ht\n\n/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\nlemma condEntropy_eq_kernel_entropy\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] [FiniteRange Y] :\n H[X | Y ; \u03bc] = Hk[condDistrib X Y \u03bc, \u03bc.map Y] := by\n rw [condEntropy_def, kernel.entropy]\n apply integral_congr_finiteSupport\n intro t ht\n simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top (\u03bc.map Y), or_false] at ht\n rw [Measure.map_apply hY (measurableSet_singleton _)] at ht\n simp only [entropy_def]\n congr\n ext S hS\n rw [condDistrib_apply' hX hY _ _ ht hS, Measure.map_apply hX hS,\n cond_apply _ (hY (measurableSet_singleton _))]\n\n/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\nlemma map_prod_comap_swap (hX : Measurable X) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9))).comap Prod.swap = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext s hs\n rw [Measure.map_apply (hZ.prod_mk hX) hs, Measure.comap_apply _ Prod.swap_injective _ _ hs]\n \u00b7 rw [Measure.map_apply (hX.prod_mk hZ)]\n \u00b7 congr with \u03c9\n simp only [Set.image_swap_eq_preimage_swap, Set.mem_preimage, Prod.swap_prod_mk]\n \u00b7 exact MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' hs\n \u00b7 exact fun t ht \u21a6 MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' ht\n\nlemma condEntropy_two_eq_kernel_entropy (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] =\n Hk[kernel.condKernel (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc),\n Measure.map Z \u03bc \u2297\u2098 kernel.fst (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc)] := by\n rw [Measure.compProd_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc),\n map_compProd_condDistrib hY hZ,\n kernel.entropy_congr (condKernel_condDistrib_ae_eq hY hX hZ \u03bc),\n \u2190 kernel.entropy_congr (swap_condDistrib_ae_eq hY hX hZ \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, Y \u03c9)) = (\u03bc.map (fun \u03c9 \u21a6 (Y \u03c9, Z \u03c9))).comap Prod.swap := by\n rw [map_prod_comap_swap hY hZ]\n rw [this, condEntropy_eq_kernel_entropy hX (hY.prod_mk hZ), kernel.entropy_comap_swap]\n\n/-- Any random variable on a zero measure space has zero conditional entropy. -/\n@[simp]\nlemma condEntropy_zero_measure (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) : H[X | Y ; (0 : Measure \u03a9)] = 0 :=\n by simp [condEntropy]\n\n/-- Conditional entropy is non-negative. -/\nlemma condEntropy_nonneg (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) : 0 \u2264 H[X | Y ; \u03bc] :=\n integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\nlemma condEntropy_le_log_card [MeasurableSingletonClass S] [Fintype S]\n (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n H[X | Y ; \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_nonneg _ _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_le_log_card _ _)\n \u00b7 have : IsProbabilityMeasure (\u03bc.map Y) := isProbabilityMeasure_map hY.aemeasurable\n simp\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY: Measurable Y) [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 y in FiniteRange.toFinset Y, (\u03bc.map Y {y}).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum' (s := FiniteRange.toFinset Y) _ _]\n simp_rw [smul_eq_mul]\n exact full_measure_of_finiteRange hY\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum_fintype\n [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY : Measurable Y) [Fintype T] :\n H[X | Y ; \u03bc] = \u2211 y, (\u03bc (Y \u207b\u00b9' {y})).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum]\n simp_rw [smul_eq_mul, Measure.map_apply hY (measurableSet_singleton _)]\n\nlemma condEntropy_prod_eq_sum {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 T'} [MeasurableSpace T']\n [MeasurableSingletonClass T']\n (\u03bc : Measure \u03a9) (hY : Measurable Y) (hZ : Measurable Z)\n [IsFiniteMeasure \u03bc] [Fintype T] [Fintype T'] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc]\n = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc[|Z \u207b\u00b9' {z}]] := by\n simp_rw [condEntropy_eq_sum_fintype _ _ _ (hY.prod_mk hZ), condEntropy_eq_sum_fintype _ _ _ hY,\n Fintype.sum_prod_type_right, Finset.mul_sum, \u2190 mul_assoc, \u2190 ENNReal.toReal_mul]\n congr with y\n congr with x\n have A : (fun a \u21a6 (Y a, Z a)) \u207b\u00b9' {(x, y)} = Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x} := by\n ext p; simp [and_comm]\n congr 2\n \u00b7 rw [cond_apply _ (hZ (measurableSet_singleton y)), \u2190 mul_assoc, A]\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {y})) 0 with hy|hy\n \u00b7 have : \u03bc (Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x}) = 0 :=\n le_antisymm ((measure_mono (Set.inter_subset_left _ _)).trans hy.le) bot_le\n simp [this, hy]\n \u00b7 rw [ENNReal.mul_inv_cancel hy (by finiteness), one_mul]\n \u00b7 rw [A, cond_cond_eq_cond_inter _ (hZ (.singleton y)) (hY (.singleton x))]\n\n/-- $H[X|Y] = \\sum_y \\sum_x P[Y=y] P[X=x|Y=y] log \\frac{1}{P[X=x|Y=y]}$.-/\nlemma condEntropy_eq_sum_sum [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T} (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc]\n = \u2211 y in FiniteRange.toFinset Y, \u2211 x in FiniteRange.toFinset X, (\u03bc.map Y {y}).toReal * negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n rw [condEntropy_eq_sum _ _ _ hY]\n congr with y\n rw [entropy_cond_eq_sum_finiteRange hX, Finset.mul_sum]\n\n/-- Same as previous lemma, but with a sum over a product space rather than a double sum. -/\nlemma condEntropy_eq_sum_prod [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T}\n (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 p in (FiniteRange.toFinset X) \u00d7\u02e2 (FiniteRange.toFinset Y),\n (\u03bc.map Y {p.2}).toReal * negMulLog ((\u03bc[|Y \u207b\u00b9' {p.2}]).map X {p.1}).toReal := by\n rw [condEntropy_eq_sum_sum hX hY, Finset.sum_product_right]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$ are random variables, and $f : T \\times S \u2192 U$ is\n injective for each fixed $t \\in T$, then $H[f(Y, X)|Y] = H[X|Y]$.\n Thus for instance $H[X-Y|Y] = H[X|Y]$. -/\nlemma condEntropy_of_injective\n [MeasurableSingletonClass U] (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (hX : Measurable X)\n (hY : Measurable Y) (f : T \u2192 S \u2192 U) (hf : \u2200 t, Injective (f t)) [FiniteRange Y] :\n H[(fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)) | Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_eq_sum _ _ _ hY, condEntropy_eq_sum _ _ _ hY]\n have : \u2200 y, H[fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)|Y\u2190 y; \u03bc] = H[(f y \u2218 X) | Y \u2190 y ; \u03bc] := by\n intro y\n refine entropy_congr ?_\n have : \u2200\u1d50 \u03c9 \u2202\u03bc[|Y \u2190 y], Y \u03c9 = y := by\n rw [ae_iff, cond_apply _ (hY (measurableSet_singleton _))]\n have : {a | \u00acY a = y} = (Y \u207b\u00b9' {y})\u1d9c := by ext; simp\n rw [this, Set.inter_compl_self, measure_empty, mul_zero]\n filter_upwards [this] with \u03c9 h\u03c9\n rw [h\u03c9]\n simp\n simp_rw [this]\n congr with y\n rw [entropy_comp_of_injective _ hX (f y) (hf y)]\n\n/-- A weaker version of the above lemma in which f is independent of Y. -/\nlemma condEntropy_comp_of_injective [MeasurableSingletonClass S] [MeasurableSingletonClass U]\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 U) (hf : Injective f) :\n H[f \u2218 X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n integral_congr_ae (ae_of_all _ (fun _ \u21a6 entropy_comp_of_injective _ hX f hf))\n\n/-- $H[X, Y| Z] = H[Y, X| Z]$. -/\nlemma condEntropy_comm {Z : \u03a9 \u2192 U}\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[\u27e8Y, X\u27e9 | Z; \u03bc] := by\n change H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Prod.swap \u2218 \u27e8X, Y\u27e9 | Z; \u03bc]\n exact (condEntropy_comp_of_injective \u03bc (hX.prod_mk hY) Prod.swap Prod.swap_injective).symm\n\nend condEntropy\n\nsection pair\n\n", "theoremStatement": "/-- One form of the chain rule : $H[X, Y] = H[X] + H[Y|X]. -/\nlemma chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y]:\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y | X ; \u03bc] ", "theoremName": "ProbabilityTheory.chain_rule'", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "a9566398a86c6e664bfcfee4a998e6630b070738", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": 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"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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.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.Lemma", "Mathlib.Tactic.TypeStar", "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.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.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.Init.Data.Int.Basic", "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.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "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.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.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.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "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.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.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n rw [entropy_eq_kernel_entropy, kernel.chain_rule]\n simp_rw [\u2190 kernel.map_const _ (hX.prod_mk hY), kernel.fst_map_prod _ hX hY, kernel.map_const _ hX,\n kernel.map_const _ (hX.prod_mk hY)]\n . congr 1\n \u00b7 rw [kernel.entropy, integral_dirac]\n rfl\n \u00b7 simp_rw [condEntropy_eq_kernel_entropy hY hX]\n have : Measure.dirac () \u2297\u2098 kernel.const Unit (\u03bc.map X) = \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) := by\n ext s _\n rw [Measure.dirac_unit_compProd_const, Measure.map_map measurable_prod_mk_left hX]\n congr\n rw [this, kernel.entropy_congr (condDistrib_const_unit hX hY \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) = (\u03bc.map X).map (Prod.mk ()) := by\n ext s _\n rw [Measure.map_map measurable_prod_mk_left hX]\n rfl\n rw [this, kernel.entropy_prodMkLeft_unit]\n \u00b7 apply kernel.FiniteKernelSupport.aefiniteKernelSupport\n exact kernel.finiteKernelSupport_of_const _", "proofType": "tactic", "proofLengthLines": 22, "proofLengthTokens": 1106}} +{"srcContext": "import PFR.FirstEstimate\nimport PFR.SecondEstimate\nimport PFR.ForMathlib.Elementary\n\n/-!\n# Endgame\n\nThe endgame on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ be independent random variables, with $X_1,\\tilde X_1$ copies\n of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $d[X_1;X_2] = k$\n* $U := X_1 + X_2$\n* $V := \\tilde X_1 + X_2$\n* $W := X_1 + \\tilde X_1$\n* $S := X_1 + X_2 + \\tilde X_1 + \\tilde X_2$.\n* $I_1 := I[U : V | S]$\n* $I_2 := I[U : W | S]$\n* $I_3 := I[V : W | S]$ (not explicitly defined in Lean)\n\n# Main results:\n\n* `sum_condMutual_le` : An upper bound on the total conditional mutual information $I_1+I_2+I_3$.\n* `sum_dist_diff_le`: A sum of the \"costs\" of $U$, $V$, $W$.\n* `construct_good`: A construction of two random variables with small Ruzsa distance between them\n given some random variables with control on total cost, as well as total mutual information.\n-/\n\nopen MeasureTheory ProbabilityTheory\nopen scoped BigOperators\n\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [m\u03a9 : MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2081', X\u2082'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[U : V | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `V = X\u2081' + X\u2082`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[U : V | S]\n\n/-- `I\u2082 := I[U : W | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `W = X\u2081' + X\u2081`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[U : W | S]\n\n--(Mantas) this times out in the proof below\nprivate lemma hmeas2 :\n Measurable fun p : Fin 4 \u2192 G => ((p 0 + p 1, p 0 + p 2), p 0 + p 1 + p 2 + p 3) := by\n apply Measurable.prod\n \u00b7 apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n\n/-- The quantity $I_3 = I[V:W|S]$ is equal to $I_2$. -/\nlemma I\u2083_eq : I[V : W | S] = I\u2082 := by\n have h_indep2 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081, X\u2082'] := by\n exact h_indep.reindex_four_cbad\n have hident : IdentDistrib (fun a (i : Fin 4) => ![X\u2081, X\u2082, X\u2081', X\u2082'] i a)\n (fun a (j : Fin 4) => ![X\u2081', X\u2082, X\u2081, X\u2082'] j a) := by\n exact { aemeasurable_fst := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n aemeasurable_snd := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n map_eq := by\n rw [\u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081, X\u2082, X\u2081', X\u2082'])\n (Fin.cases hX\u2081 <| Fin.cases hX\u2082 <| Fin.cases hX\u2081' <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep,\n \u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081', X\u2082, X\u2081, X\u2082'])\n (Fin.cases hX\u2081' <| Fin.cases hX\u2082 <| Fin.cases hX\u2081 <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep2]\n congr\n ext i\n fin_cases i\n all_goals simp [h\u2081.map_eq] }\n have hmeas1 : Measurable (fun p : Fin 4 \u2192 G => (p 0 + p 1, p 0 + p 1 + p 2 + p 3)) := by\n simp_all only [Matrix.cons_val', Matrix.empty_val', Matrix.cons_val_fin_one]\n apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n have hUVS : IdentDistrib (prod U S) (prod V S) := by\n convert (IdentDistrib.comp hident hmeas1); simp; abel\n have hUVWS : IdentDistrib (prod (prod U W) S) (prod (prod V W) S) := by\n convert (IdentDistrib.comp hident hmeas2) <;> simp <;> abel\n have hU : Measurable U := Measurable.add hX\u2081 hX\u2082\n have hV : Measurable V := Measurable.add hX\u2081' hX\u2082\n have hW : Measurable W := Measurable.add hX\u2081' hX\u2081\n have hS : Measurable S := by measurability\n rw [condMutualInfo_eq hV hW hS, condMutualInfo_eq hU hW hS, chain_rule'' \u2119 hU hS,\n chain_rule'' \u2119 hV hS, chain_rule'' \u2119 hW hS, chain_rule'' \u2119 _ hS, chain_rule'' \u2119 _ hS,\n IdentDistrib.entropy_eq hUVS, IdentDistrib.entropy_eq hUVWS]\n \u00b7 exact Measurable.prod (by exact hU) (by exact hW)\n \u00b7 exact Measurable.prod (by exact hV) (by exact hW)\n\n\n/--\n$$ I(U : V | S) + I(V : W | S) + I(W : U | S) $$\nis less than or equal to\n$$ 6 \\eta k - \\frac{1 - 5 \\eta}{1-\\eta} (2 \\eta k - I_1).$$\n-/\nlemma sum_condMutual_le :\n I[U : V | S] + I[V : W | S] + I[W : U | S]\n \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) := by\n have : I[W:U|S] = I\u2082 := by\n rw [condMutualInfo_comm]\n \u00b7 exact Measurable.add' hX\u2081' hX\u2081\n \u00b7 exact Measurable.add' hX\u2081 hX\u2082\n rw [I\u2083_eq, this]\n have h\u2082 := second_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n have h := add_le_add (add_le_add_left h\u2082 I\u2081) h\u2082\n convert h using 1\n have : 1 - p.\u03b7 > 0 := by linarith [p.h\u03b7']\n field_simp [this]\n ring\n all_goals { simpa }\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" => d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nlemma hU : H[U] = H[X\u2081' + X\u2082'] :=\n IdentDistrib.entropy_eq (ProbabilityTheory.IdentDistrib.add h\u2081 h\u2082\n (iIndepFun.indepFun h_indep (show (0 : Fin 4) \u2260 1 by norm_cast))\n (iIndepFun.indepFun h_indep (show (2 : Fin 4) \u2260 3 by norm_cast)))\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition1 : iIndepFun (fun _ \u21a6 hG) ![X\u2081, X\u2082, X\u2081' + X\u2082'] :=\n h_indep.apply_two_last hX\u2081 hX\u2082 hX\u2081' hX\u2082' measurable_add\n\nlemma hV : H[V] = H[X\u2081 + X\u2082'] :=\nIdentDistrib.entropy_eq (ProbabilityTheory.IdentDistrib.add h\u2081.symm h\u2082\n (iIndepFun.indepFun h_indep (show (2 : Fin 4) \u2260 1 by norm_cast))\n (iIndepFun.indepFun h_indep (show (0 : Fin 4) \u2260 3 by norm_cast)))\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition2 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2081, X\u2081' + X\u2082'] :=\n independenceCondition1 hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_bacd\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition3 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081 + X\u2082'] :=\n independenceCondition1 hX\u2081' hX\u2082 hX\u2081 hX\u2082' h_indep.reindex_four_cbad\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition4 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2081', X\u2081 + X\u2082'] :=\n independenceCondition1 hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bcad\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition5 : iIndepFun (fun _ \u21a6 hG) ![X\u2081, X\u2081', X\u2082 + X\u2082'] :=\n independenceCondition1 hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_acbd\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition6 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2082', X\u2081' + X\u2081] :=\n independenceCondition1 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bdca\n\nset_option maxHeartbeats 400000 in\n/--\n$$ \\sum_{i=1}^2 \\sum_{A\\in\\{U,V,W\\}} \\big(d[X^0_i;A|S] - d[X^0_i;X_i]\\big)$$\nis less than or equal to\n$$ \\leq (6 - 3\\eta) k + 3(2 \\eta k - I_1).$$\n-/\nlemma sum_dist_diff_le :\n c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 (6 - 3 * p.\u03b7)*k + 3 * (2*p.\u03b7*k - I\u2081) := by\n let X\u2080\u2081 := p.X\u2080\u2081\n let X\u2080\u2082 := p.X\u2080\u2082\n have ineq1 : d[X\u2080\u2081 # U | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 := by\n have aux1 : H[S] + H[U] - H[X\u2081] - H[X\u2081' + X\u2082'] = H[S] - H[X\u2081] := by\n rw [hU X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep]; ring\n have aux2 : d[X\u2080\u2081 # U | U + (X\u2081' + X\u2082')] - d[X\u2080\u2081 # X\u2081]\n \u2264 (H[U + (X\u2081' + X\u2082')] + H[U] - H[X\u2081] - H[X\u2081' + X\u2082']) / 2 :=\n condRuzsaDist_diff_ofsum_le \u2119 (hX := p.hmeas1) (hY := hX\u2081) (hZ := hX\u2082)\n (Measurable.add hX\u2081' hX\u2082') (independenceCondition1 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n rw [\u2190 add_assoc, aux1] at aux2\n linarith [aux2]\n\n have ineq2 : d[X\u2080\u2082 # U | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := by\n have aux1 : H[S] + H[U] - H[X\u2082] - H[X\u2081' + X\u2082'] = H[S] - H[X\u2082] := by\n rw [hU X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep] ; ring\n have aux2 : d[X\u2080\u2082 # U | U + (X\u2081' + X\u2082')] - d[X\u2080\u2082 # X\u2082]\n \u2264 (H[U + (X\u2081' + X\u2082')] + H[U] - H[X\u2082] - H[X\u2081' + X\u2082']) / 2 := by\n rw [(show U = X\u2082 + X\u2081 from add_comm _ _)]\n apply condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas2) (hX\u2082) (hX\u2081)\n (Measurable.add hX\u2081' hX\u2082') (independenceCondition2 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n rw [\u2190 add_assoc, aux1] at aux2\n linarith [aux2]\n\n have V_add_eq : V + (X\u2081 + X\u2082') = S := by abel\n\n have ineq3 : d[X\u2080\u2081 # V | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 := by\n have aux2 : d[p.X\u2080\u2081 # V | V + (X\u2081 + X\u2082')] - d[p.X\u2080\u2081 # X\u2081']\n \u2264 (H[V + (X\u2081 + X\u2082')] + H[V] - H[X\u2081'] - H[X\u2081 + X\u2082']) / 2 :=\n condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas1) (hX\u2081') (hX\u2082) (Measurable.add hX\u2081 hX\u2082')\n (independenceCondition3 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n\n have aux1 : H[S] + H[V] - H[X\u2081'] - H[X\u2081 + X\u2082'] = H[S ; \u2119] - H[X\u2081 ; \u2119] := by\n rw [hV X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep, h\u2081.entropy_eq]; ring\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl p.hmeas1.aemeasurable) h\u2081,\n V_add_eq, aux1] at aux2\n linarith [aux2]\n\n have ineq4 : d[X\u2080\u2082 # V | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := by\n have aux2 : d[p.X\u2080\u2082 # V | V + (X\u2081 + X\u2082')] - d[p.X\u2080\u2082 # X\u2082]\n \u2264 (H[V + (X\u2081 + X\u2082')] + H[V] - H[X\u2082] - H[X\u2081 + X\u2082']) / 2 := by\n rw [(show V = X\u2082 + X\u2081' from add_comm _ _)]\n apply condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas2) (hX\u2082) (hX\u2081') (Measurable.add hX\u2081 hX\u2082')\n (independenceCondition4 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have aux1 : H[S] + H[V] - H[X\u2082] - H[X\u2081 + X\u2082'] = H[S ; \u2119] - H[X\u2082 ; \u2119] := by\n rw [hV X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep]; ring\n rw [V_add_eq, aux1] at aux2\n linarith [aux2]\n\n let W' := X\u2082 + X\u2082'\n have ineq5 : d[X\u2080\u2081 # W | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] + H[W ; \u2119] - H[X\u2081 ; \u2119] - H[W' ; \u2119])/2 := by\n have := condRuzsaDist_diff_ofsum_le \u2119 p.hmeas1 hX\u2081 hX\u2081' (Measurable.add hX\u2082 hX\u2082')\n (independenceCondition5 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have S_eq : X\u2081 + X\u2081' + (fun a \u21a6 X\u2082 a + X\u2082' a) = S := by\n rw [(show (fun a \u21a6 X\u2082 a + X\u2082' a) = X\u2082 + X\u2082' by rfl), \u2190 add_assoc, add_assoc X\u2081, add_comm X\u2081',\n \u2190 add_assoc]\n rwa [S_eq, add_comm X\u2081 X\u2081'] at this\n\n have ineq6 : d[X\u2080\u2082 # W' | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] + H[W' ; \u2119] - H[X\u2082 ; \u2119] - H[W ; \u2119])/2 := by\n have := condRuzsaDist_diff_ofsum_le \u2119 p.hmeas2 hX\u2082 hX\u2082' (Measurable.add hX\u2081' hX\u2081)\n (independenceCondition6 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have S_eq : X\u2082 + X\u2082' + (fun a \u21a6 X\u2081' a + X\u2081 a) = S := by\n rw [(show (fun a \u21a6 X\u2081' a + X\u2081 a) = X\u2081' + X\u2081 by rfl), add_comm, \u2190 add_assoc, add_comm X\u2081',\n add_assoc X\u2081, add_comm X\u2081', \u2190 add_assoc]\n rwa [S_eq] at this\n\n have dist_eq : d[X\u2080\u2082 # W' | S] = d[X\u2080\u2082 # W | S] := by\n have S_eq : S = (X\u2082 + X\u2082') + (X\u2081' + X\u2081) := by\n rw [add_comm X\u2081' X\u2081, add_assoc _ X\u2082', add_comm X\u2082', \u2190 add_assoc X\u2082, \u2190 add_assoc X\u2082,\n add_comm X\u2082]\n rw [S_eq]\n apply condRuzsaDist'_of_inj_map' p.hmeas2 (hX\u2082.add hX\u2082') (hX\u2081'.add hX\u2081)\n\n -- Put everything together to bound the sum of the `c` terms\n have ineq7 : c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264\n 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := by\n have step\u2081 : c[U|S # U|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc\n _ = (d[p.X\u2080\u2081 # U|S] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2082 # U|S] - d[p.X\u2080\u2082 # X\u2082]) := by ring\n _ \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 + (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := add_le_add ineq1 ineq2\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n have step\u2082 : c[V|S # V|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc c[V|S # V|S] =(d[p.X\u2080\u2081 # V|S] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2082 # V|S] - d[p.X\u2080\u2082 # X\u2082]) := by ring\n _ \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 + (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := add_le_add ineq3 ineq4\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n have step\u2083 : c[W|S # W|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc c[W|S # W|S] = (d[X\u2080\u2081 # W | S] - d[X\u2080\u2081 # X\u2081]) + (d[X\u2080\u2082 # W' | S] - d[X\u2080\u2082 # X\u2082]) :=\n by rw [dist_eq]\n _ \u2264 (H[S ; \u2119] + H[W ; \u2119] - H[X\u2081 ; \u2119] - H[W' ; \u2119])/2 + (H[S ; \u2119] + H[W' ; \u2119] - H[X\u2082 ; \u2119] - H[W ; \u2119])/2\n := add_le_add ineq5 ineq6\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) +\n (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) + (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) :=\n add_le_add (add_le_add step\u2081 step\u2082) step\u2083\n _ = 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := by ring\n\n have h_indep' : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'] := by\n apply ProbabilityTheory.iIndepFun.reindex (Equiv.swap (2 : Fin 4) 3)\n convert h_indep using 1\n ext x\n fin_cases x ; all_goals { aesop }\n\n have ineq8 : 3 * H[S ; \u2119] \u2264 3/2 * (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081 :=\n calc 3 * H[S ; \u2119] \u2264 3 * (H[X\u2081 ; \u2119] / 2 + H[X\u2082 ; \u2119] / 2 + (2+p.\u03b7)*k - I\u2081) := by\n apply (mul_le_mul_left (zero_lt_three' \u211d)).mpr\n (ent_ofsum_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep' h_min)\n _ = 3/2 * ( H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081 := by ring\n\n -- Final computation\n calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := ineq7\n _ = 3 * H[S ; \u2119] - (3/2 *(H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])) := by ring\n _ \u2264 (3/2 * ( H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081) - (3/2 *(H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])) :=\n sub_le_sub_right ineq8 _\n _ = (6 - 3 * p.\u03b7)*k + 3 * (2*p.\u03b7*k - I\u2081) := by ring\n\n/-- $U+V+W=0$. -/\nlemma sum_uvw_eq_zero : U+V+W = 0 := by\n rw [add_comm X\u2081' X\u2082, ElementaryAddCommGroup.sum_add_sum_add_sum_eq_zero]\n\nsection construct_good\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081+T\u2082+T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\nThen there exist random variables $T'_1, T'_2$ such that\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X_1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2]) $$\nis at most\n$$ \\delta + \\eta ( d[X^0_1;T_1]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2]-d[X^0_2;X_2]) $$\n$$ + \\tfrac12 \\eta I[T_1: T_3] + \\tfrac12 \\eta I[T_2: T_3].$$\n-/\nlemma construct_good_prelim :\n k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 # T\u2082] + p.\u03b7 * (I[T\u2081: T\u2083] + I[T\u2082 : T\u2083])/2 := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n\n have hp.\u03b7 : 0 \u2264 p.\u03b7 := by linarith [p.h\u03b7]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 :=\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have h2T\u2081 : T\u2081 = T\u2082 + T\u2083 := by simp [h2T\u2083, add_left_comm]\n have h2T\u2082 : T\u2082 = T\u2083 + T\u2081 := by simp [h2T\u2081, add_left_comm]\n\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n\n have h2 : p.\u03b7 * sum2 \u2264 p.\u03b7 * (d[p.X\u2080\u2081 # T\u2081] - d[p.X\u2080\u2081 # X\u2081] + I[T\u2081 : T\u2083] / 2) := by\n have : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), sum2]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n gcongr\n linarith [condRuzsaDist_le' \u2119 \u2119 p.hmeas1 hT\u2081 hT\u2083]\n\n have h3 : p.\u03b7 * sum3 \u2264 p.\u03b7 * (d[p.X\u2080\u2082 # T\u2082] - d[p.X\u2080\u2082 # X\u2082] + I[T\u2082 : T\u2083] / 2) := by\n have : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp [sum3, integral_sub (.of_finite _ _) (.of_finite _ _)]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n gcongr\n linarith [condRuzsaDist_le' \u2119 \u2119 p.hmeas2 hT\u2082 hT\u2083]\n\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * c[T\u2081 # T\u2082] + p.\u03b7 * (I[T\u2081 : T\u2083] + I[T\u2082 : T\u2083]) / 2 := by\n suffices sum4 = sum1 + p.\u03b7 * (sum2 + sum3) by linarith\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _), integral_mul_left]\n\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $ ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\n/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n-\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\n\nThen there exist random variables $T'_1, T'_2$ such that\n\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X _1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2])$$\n\nis at most\n\n$$\\delta + \\frac{\\eta}{3} \\biggl( \\delta + \\sum_{i=1}^2 \\sum_{j = 1}^3\n (d[X^0_i;T_j] - d[X^0_i; X_i]) \\biggr).$$\n-/\nlemma construct_good : k \u2264 \u03b4 + (p.\u03b7/3) * (\u03b4 + c[T\u2081 # T\u2081] + c[T\u2082 # T\u2082] + c[T\u2083 # T\u2083]) := by\n have v2 := construct_good_prelim p X\u2081 X\u2082 h_min (by rw [\u2190 hT]; abel) hT\u2081 hT\u2083 hT\u2082\n have v3 := construct_good_prelim p X\u2081 X\u2082 h_min (by rw [\u2190 hT]; abel) hT\u2082 hT\u2081 hT\u2083\n have v6 := construct_good_prelim p X\u2081 X\u2082 h_min (by rw [\u2190 hT]; abel) hT\u2083 hT\u2082 hT\u2081\n simp only [mutualInfo, entropy_comm hT\u2082 hT\u2081, entropy_comm hT\u2083 hT\u2081, entropy_comm hT\u2083 hT\u2082]\n at *\n linarith\n\nlemma construct_good' (\u03bc : Measure \u03a9') [IsProbabilityMeasure \u03bc]:\n k \u2264 \u03b4[\u03bc] + (p.\u03b7/3) * (\u03b4[\u03bc] + c[T\u2081 ; \u03bc # T\u2081 ; \u03bc] + c[T\u2082 ; \u03bc # T\u2082 ; \u03bc] + c[T\u2083 ; \u03bc # T\u2083 ; \u03bc]) := by\n letI : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n apply construct_good p X\u2081 X\u2082 h_min hT hT\u2081 hT\u2082 hT\u2083\n\nlemma cond_c_eq_integral {Y Z : \u03a9' \u2192 G} (hY : Measurable Y) (hZ : Measurable Z) : c[Y | Z # Y | Z] =\n (Measure.map Z \u2119)[fun z => c[Y ; \u2119[|Z \u2190 z] # Y ; \u2119[|Z \u2190 z]]] := by\n simp only [integral_eq_sum, smul_sub, smul_add, smul_sub, Finset.sum_sub_distrib,\n Finset.sum_add_distrib]\n simp_rw [\u2190 integral_eq_sum]\n rw [\u2190 condRuzsaDist'_eq_integral _ hY hZ, \u2190 condRuzsaDist'_eq_integral _ hY hZ, integral_const,\n integral_const]\n have : IsProbabilityMeasure (Measure.map Z \u2119) := isProbabilityMeasure_map hZ.aemeasurable\n simp only [measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul]\n\nvariable {R : \u03a9' \u2192 G} (hR : Measurable R)\nlocal notation3:max \"\u03b4'\" => I[T\u2081 : T\u2082|R] + I[T\u2082 : T\u2083|R] + I[T\u2083 : T\u2081|R]\n\nlemma delta'_eq_integral : \u03b4' = (Measure.map R \u2119)[fun r => \u03b4[\u2119[|R\u207b\u00b9' {r}]]] := by\n simp_rw [condMutualInfo_eq_integral_mutualInfo, integral_eq_sum, smul_add,\n Finset.sum_add_distrib]\n\n", "theoremStatement": "lemma cond_construct_good :\n k \u2264 \u03b4' + (p.\u03b7/3) * (\u03b4' + c[T\u2081 | R # T\u2081 | R] + c[T\u2082 | R # T\u2082 | R] + c[T\u2083 | R # T\u2083 | R]) ", "theoremName": "cond_construct_good", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "fafcc9e942a4844a347223ca0e5ade8d7f76bfb5", "date": "2023-11-27"}, "file": "PFR/PFR/Endgame.lean", "module": "PFR.Endgame", "jsonFile": 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"Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [delta'_eq_integral, cond_c_eq_integral _ _ _ hT\u2081 hR, cond_c_eq_integral _ _ _ hT\u2082 hR,\n cond_c_eq_integral _ _ _ hT\u2083 hR]\n simp_rw [integral_eq_sum, \u2190 Finset.sum_add_distrib, \u2190 smul_add, Finset.mul_sum, mul_smul_comm,\n \u2190 Finset.sum_add_distrib, \u2190 smul_add]\n simp_rw [\u2190 integral_eq_sum]\n have : IsProbabilityMeasure (Measure.map R \u2119) := isProbabilityMeasure_map (by measurability)\n calc\n k = (Measure.map R \u2119)[fun _r => k] := by\n rw [integral_const]; simp\n _ \u2264 _ := ?_\n simp_rw [integral_eq_sum]\n apply Finset.sum_le_sum\n intro r _\n by_cases hr : \u2119 (R\u207b\u00b9' {r}) = 0\n \u00b7 rw [Measure.map_apply hR (MeasurableSet.singleton r), hr]\n simp\n simp_rw [smul_eq_mul]\n gcongr (?_ * ?_)\n \u00b7 apply rdist_nonneg hX\u2081 hX\u2082\n \u00b7 rfl\n have : IsProbabilityMeasure (\u2119[|R \u207b\u00b9' {r}]) := cond_isProbabilityMeasure \u2119 hr\n apply construct_good' p X\u2081 X\u2082 h_min hT hT\u2081 hT\u2082 hT\u2083", "proofType": "tactic", "proofLengthLines": 22, "proofLengthTokens": 883}} +{"srcContext": "import PFR.FirstEstimate\nimport PFR.SecondEstimate\nimport PFR.ForMathlib.Elementary\n\n/-!\n# Endgame\n\nThe endgame on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ be independent random variables, with $X_1,\\tilde X_1$ copies\n of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $d[X_1;X_2] = k$\n* $U := X_1 + X_2$\n* $V := \\tilde X_1 + X_2$\n* $W := X_1 + \\tilde X_1$\n* $S := X_1 + X_2 + \\tilde X_1 + \\tilde X_2$.\n* $I_1 := I[U : V | S]$\n* $I_2 := I[U : W | S]$\n* $I_3 := I[V : W | S]$ (not explicitly defined in Lean)\n\n# Main results:\n\n* `sum_condMutual_le` : An upper bound on the total conditional mutual information $I_1+I_2+I_3$.\n* `sum_dist_diff_le`: A sum of the \"costs\" of $U$, $V$, $W$.\n* `construct_good`: A construction of two random variables with small Ruzsa distance between them\n given some random variables with control on total cost, as well as total mutual information.\n-/\n\nopen MeasureTheory ProbabilityTheory\nopen scoped BigOperators\n\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [m\u03a9 : MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2081', X\u2082'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[U : V | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `V = X\u2081' + X\u2082`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[U : V | S]\n\n/-- `I\u2082 := I[U : W | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `W = X\u2081' + X\u2081`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[U : W | S]\n\n--(Mantas) this times out in the proof below\nprivate lemma hmeas2 :\n Measurable fun p : Fin 4 \u2192 G => ((p 0 + p 1, p 0 + p 2), p 0 + p 1 + p 2 + p 3) := by\n apply Measurable.prod\n \u00b7 apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n\n/-- The quantity $I_3 = I[V:W|S]$ is equal to $I_2$. -/\nlemma I\u2083_eq : I[V : W | S] = I\u2082 := by\n have h_indep2 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081, X\u2082'] := by\n exact h_indep.reindex_four_cbad\n have hident : IdentDistrib (fun a (i : Fin 4) => ![X\u2081, X\u2082, X\u2081', X\u2082'] i a)\n (fun a (j : Fin 4) => ![X\u2081', X\u2082, X\u2081, X\u2082'] j a) := by\n exact { aemeasurable_fst := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n aemeasurable_snd := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n map_eq := by\n rw [\u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081, X\u2082, X\u2081', X\u2082'])\n (Fin.cases hX\u2081 <| Fin.cases hX\u2082 <| Fin.cases hX\u2081' <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep,\n \u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081', X\u2082, X\u2081, X\u2082'])\n (Fin.cases hX\u2081' <| Fin.cases hX\u2082 <| Fin.cases hX\u2081 <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep2]\n congr\n ext i\n fin_cases i\n all_goals simp [h\u2081.map_eq] }\n have hmeas1 : Measurable (fun p : Fin 4 \u2192 G => (p 0 + p 1, p 0 + p 1 + p 2 + p 3)) := by\n simp_all only [Matrix.cons_val', Matrix.empty_val', Matrix.cons_val_fin_one]\n apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n have hUVS : IdentDistrib (prod U S) (prod V S) := by\n convert (IdentDistrib.comp hident hmeas1); simp; abel\n have hUVWS : IdentDistrib (prod (prod U W) S) (prod (prod V W) S) := by\n convert (IdentDistrib.comp hident hmeas2) <;> simp <;> abel\n have hU : Measurable U := Measurable.add hX\u2081 hX\u2082\n have hV : Measurable V := Measurable.add hX\u2081' hX\u2082\n have hW : Measurable W := Measurable.add hX\u2081' hX\u2081\n have hS : Measurable S := by measurability\n rw [condMutualInfo_eq hV hW hS, condMutualInfo_eq hU hW hS, chain_rule'' \u2119 hU hS,\n chain_rule'' \u2119 hV hS, chain_rule'' \u2119 hW hS, chain_rule'' \u2119 _ hS, chain_rule'' \u2119 _ hS,\n IdentDistrib.entropy_eq hUVS, IdentDistrib.entropy_eq hUVWS]\n \u00b7 exact Measurable.prod (by exact hU) (by exact hW)\n \u00b7 exact Measurable.prod (by exact hV) (by exact hW)\n\n\n/--\n$$ I(U : V | S) + I(V : W | S) + I(W : U | S) $$\nis less than or equal to\n$$ 6 \\eta k - \\frac{1 - 5 \\eta}{1-\\eta} (2 \\eta k - I_1).$$\n-/\nlemma sum_condMutual_le :\n I[U : V | S] + I[V : W | S] + I[W : U | S]\n \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) := by\n have : I[W:U|S] = I\u2082 := by\n rw [condMutualInfo_comm]\n \u00b7 exact Measurable.add' hX\u2081' hX\u2081\n \u00b7 exact Measurable.add' hX\u2081 hX\u2082\n rw [I\u2083_eq, this]\n have h\u2082 := second_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n have h := add_le_add (add_le_add_left h\u2082 I\u2081) h\u2082\n convert h using 1\n have : 1 - p.\u03b7 > 0 := by linarith [p.h\u03b7']\n field_simp [this]\n ring\n all_goals { simpa }\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" => d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nlemma hU : H[U] = H[X\u2081' + X\u2082'] :=\n IdentDistrib.entropy_eq (ProbabilityTheory.IdentDistrib.add h\u2081 h\u2082\n (iIndepFun.indepFun h_indep (show (0 : Fin 4) \u2260 1 by norm_cast))\n (iIndepFun.indepFun h_indep (show (2 : Fin 4) \u2260 3 by norm_cast)))\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition1 : iIndepFun (fun _ \u21a6 hG) ![X\u2081, X\u2082, X\u2081' + X\u2082'] :=\n h_indep.apply_two_last hX\u2081 hX\u2082 hX\u2081' hX\u2082' measurable_add\n\nlemma hV : H[V] = H[X\u2081 + X\u2082'] :=\nIdentDistrib.entropy_eq (ProbabilityTheory.IdentDistrib.add h\u2081.symm h\u2082\n (iIndepFun.indepFun h_indep (show (2 : Fin 4) \u2260 1 by norm_cast))\n (iIndepFun.indepFun h_indep (show (0 : Fin 4) \u2260 3 by norm_cast)))\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition2 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2081, X\u2081' + X\u2082'] :=\n independenceCondition1 hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_bacd\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition3 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081 + X\u2082'] :=\n independenceCondition1 hX\u2081' hX\u2082 hX\u2081 hX\u2082' h_indep.reindex_four_cbad\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition4 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2081', X\u2081 + X\u2082'] :=\n independenceCondition1 hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bcad\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition5 : iIndepFun (fun _ \u21a6 hG) ![X\u2081, X\u2081', X\u2082 + X\u2082'] :=\n independenceCondition1 hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_acbd\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition6 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2082', X\u2081' + X\u2081] :=\n independenceCondition1 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bdca\n\nset_option maxHeartbeats 400000 in\n/--\n$$ \\sum_{i=1}^2 \\sum_{A\\in\\{U,V,W\\}} \\big(d[X^0_i;A|S] - d[X^0_i;X_i]\\big)$$\nis less than or equal to\n$$ \\leq (6 - 3\\eta) k + 3(2 \\eta k - I_1).$$\n-/\nlemma sum_dist_diff_le :\n c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 (6 - 3 * p.\u03b7)*k + 3 * (2*p.\u03b7*k - I\u2081) := by\n let X\u2080\u2081 := p.X\u2080\u2081\n let X\u2080\u2082 := p.X\u2080\u2082\n have ineq1 : d[X\u2080\u2081 # U | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 := by\n have aux1 : H[S] + H[U] - H[X\u2081] - H[X\u2081' + X\u2082'] = H[S] - H[X\u2081] := by\n rw [hU X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep]; ring\n have aux2 : d[X\u2080\u2081 # U | U + (X\u2081' + X\u2082')] - d[X\u2080\u2081 # X\u2081]\n \u2264 (H[U + (X\u2081' + X\u2082')] + H[U] - H[X\u2081] - H[X\u2081' + X\u2082']) / 2 :=\n condRuzsaDist_diff_ofsum_le \u2119 (hX := p.hmeas1) (hY := hX\u2081) (hZ := hX\u2082)\n (Measurable.add hX\u2081' hX\u2082') (independenceCondition1 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n rw [\u2190 add_assoc, aux1] at aux2\n linarith [aux2]\n\n have ineq2 : d[X\u2080\u2082 # U | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := by\n have aux1 : H[S] + H[U] - H[X\u2082] - H[X\u2081' + X\u2082'] = H[S] - H[X\u2082] := by\n rw [hU X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep] ; ring\n have aux2 : d[X\u2080\u2082 # U | U + (X\u2081' + X\u2082')] - d[X\u2080\u2082 # X\u2082]\n \u2264 (H[U + (X\u2081' + X\u2082')] + H[U] - H[X\u2082] - H[X\u2081' + X\u2082']) / 2 := by\n rw [(show U = X\u2082 + X\u2081 from add_comm _ _)]\n apply condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas2) (hX\u2082) (hX\u2081)\n (Measurable.add hX\u2081' hX\u2082') (independenceCondition2 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n rw [\u2190 add_assoc, aux1] at aux2\n linarith [aux2]\n\n have V_add_eq : V + (X\u2081 + X\u2082') = S := by abel\n\n have ineq3 : d[X\u2080\u2081 # V | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 := by\n have aux2 : d[p.X\u2080\u2081 # V | V + (X\u2081 + X\u2082')] - d[p.X\u2080\u2081 # X\u2081']\n \u2264 (H[V + (X\u2081 + X\u2082')] + H[V] - H[X\u2081'] - H[X\u2081 + X\u2082']) / 2 :=\n condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas1) (hX\u2081') (hX\u2082) (Measurable.add hX\u2081 hX\u2082')\n (independenceCondition3 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n\n have aux1 : H[S] + H[V] - H[X\u2081'] - H[X\u2081 + X\u2082'] = H[S ; \u2119] - H[X\u2081 ; \u2119] := by\n rw [hV X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep, h\u2081.entropy_eq]; ring\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl p.hmeas1.aemeasurable) h\u2081,\n V_add_eq, aux1] at aux2\n linarith [aux2]\n\n have ineq4 : d[X\u2080\u2082 # V | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := by\n have aux2 : d[p.X\u2080\u2082 # V | V + (X\u2081 + X\u2082')] - d[p.X\u2080\u2082 # X\u2082]\n \u2264 (H[V + (X\u2081 + X\u2082')] + H[V] - H[X\u2082] - H[X\u2081 + X\u2082']) / 2 := by\n rw [(show V = X\u2082 + X\u2081' from add_comm _ _)]\n apply condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas2) (hX\u2082) (hX\u2081') (Measurable.add hX\u2081 hX\u2082')\n (independenceCondition4 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have aux1 : H[S] + H[V] - H[X\u2082] - H[X\u2081 + X\u2082'] = H[S ; \u2119] - H[X\u2082 ; \u2119] := by\n rw [hV X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep]; ring\n rw [V_add_eq, aux1] at aux2\n linarith [aux2]\n\n let W' := X\u2082 + X\u2082'\n have ineq5 : d[X\u2080\u2081 # W | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] + H[W ; \u2119] - H[X\u2081 ; \u2119] - H[W' ; \u2119])/2 := by\n have := condRuzsaDist_diff_ofsum_le \u2119 p.hmeas1 hX\u2081 hX\u2081' (Measurable.add hX\u2082 hX\u2082')\n (independenceCondition5 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have S_eq : X\u2081 + X\u2081' + (fun a \u21a6 X\u2082 a + X\u2082' a) = S := by\n rw [(show (fun a \u21a6 X\u2082 a + X\u2082' a) = X\u2082 + X\u2082' by rfl), \u2190 add_assoc, add_assoc X\u2081, add_comm X\u2081',\n \u2190 add_assoc]\n rwa [S_eq, add_comm X\u2081 X\u2081'] at this\n\n have ineq6 : d[X\u2080\u2082 # W' | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] + H[W' ; \u2119] - H[X\u2082 ; \u2119] - H[W ; \u2119])/2 := by\n have := condRuzsaDist_diff_ofsum_le \u2119 p.hmeas2 hX\u2082 hX\u2082' (Measurable.add hX\u2081' hX\u2081)\n (independenceCondition6 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have S_eq : X\u2082 + X\u2082' + (fun a \u21a6 X\u2081' a + X\u2081 a) = S := by\n rw [(show (fun a \u21a6 X\u2081' a + X\u2081 a) = X\u2081' + X\u2081 by rfl), add_comm, \u2190 add_assoc, add_comm X\u2081',\n add_assoc X\u2081, add_comm X\u2081', \u2190 add_assoc]\n rwa [S_eq] at this\n\n have dist_eq : d[X\u2080\u2082 # W' | S] = d[X\u2080\u2082 # W | S] := by\n have S_eq : S = (X\u2082 + X\u2082') + (X\u2081' + X\u2081) := by\n rw [add_comm X\u2081' X\u2081, add_assoc _ X\u2082', add_comm X\u2082', \u2190 add_assoc X\u2082, \u2190 add_assoc X\u2082,\n add_comm X\u2082]\n rw [S_eq]\n apply condRuzsaDist'_of_inj_map' p.hmeas2 (hX\u2082.add hX\u2082') (hX\u2081'.add hX\u2081)\n\n -- Put everything together to bound the sum of the `c` terms\n have ineq7 : c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264\n 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := by\n have step\u2081 : c[U|S # U|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc\n _ = (d[p.X\u2080\u2081 # U|S] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2082 # U|S] - d[p.X\u2080\u2082 # X\u2082]) := by ring\n _ \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 + (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := add_le_add ineq1 ineq2\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n have step\u2082 : c[V|S # V|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc c[V|S # V|S] =(d[p.X\u2080\u2081 # V|S] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2082 # V|S] - d[p.X\u2080\u2082 # X\u2082]) := by ring\n _ \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 + (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := add_le_add ineq3 ineq4\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n have step\u2083 : c[W|S # W|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc c[W|S # W|S] = (d[X\u2080\u2081 # W | S] - d[X\u2080\u2081 # X\u2081]) + (d[X\u2080\u2082 # W' | S] - d[X\u2080\u2082 # X\u2082]) :=\n by rw [dist_eq]\n _ \u2264 (H[S ; \u2119] + H[W ; \u2119] - H[X\u2081 ; \u2119] - H[W' ; \u2119])/2 + (H[S ; \u2119] + H[W' ; \u2119] - H[X\u2082 ; \u2119] - H[W ; \u2119])/2\n := add_le_add ineq5 ineq6\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) +\n (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) + (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) :=\n add_le_add (add_le_add step\u2081 step\u2082) step\u2083\n _ = 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := by ring\n\n have h_indep' : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'] := by\n apply ProbabilityTheory.iIndepFun.reindex (Equiv.swap (2 : Fin 4) 3)\n convert h_indep using 1\n ext x\n fin_cases x ; all_goals { aesop }\n\n have ineq8 : 3 * H[S ; \u2119] \u2264 3/2 * (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081 :=\n calc 3 * H[S ; \u2119] \u2264 3 * (H[X\u2081 ; \u2119] / 2 + H[X\u2082 ; \u2119] / 2 + (2+p.\u03b7)*k - I\u2081) := by\n apply (mul_le_mul_left (zero_lt_three' \u211d)).mpr\n (ent_ofsum_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep' h_min)\n _ = 3/2 * ( H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081 := by ring\n\n -- Final computation\n calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := ineq7\n _ = 3 * H[S ; \u2119] - (3/2 *(H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])) := by ring\n _ \u2264 (3/2 * ( H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081) - (3/2 *(H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])) :=\n sub_le_sub_right ineq8 _\n _ = (6 - 3 * p.\u03b7)*k + 3 * (2*p.\u03b7*k - I\u2081) := by ring\n\n/-- $U+V+W=0$. -/\nlemma sum_uvw_eq_zero : U+V+W = 0 := by\n rw [add_comm X\u2081' X\u2082, ElementaryAddCommGroup.sum_add_sum_add_sum_eq_zero]\n\nsection construct_good\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081+T\u2082+T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\nThen there exist random variables $T'_1, T'_2$ such that\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X_1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2]) $$\nis at most\n$$ \\delta + \\eta ( d[X^0_1;T_1]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2]-d[X^0_2;X_2]) $$\n$$ + \\tfrac12 \\eta I[T_1: T_3] + \\tfrac12 \\eta I[T_2: T_3].$$\n-/\nlemma construct_good_prelim :\n k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 # T\u2082] + p.\u03b7 * (I[T\u2081: T\u2083] + I[T\u2082 : T\u2083])/2 := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n\n have hp.\u03b7 : 0 \u2264 p.\u03b7 := by linarith [p.h\u03b7]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 :=\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have h2T\u2081 : T\u2081 = T\u2082 + T\u2083 := by simp [h2T\u2083, add_left_comm]\n have h2T\u2082 : T\u2082 = T\u2083 + T\u2081 := by simp [h2T\u2081, add_left_comm]\n\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n\n have h2 : p.\u03b7 * sum2 \u2264 p.\u03b7 * (d[p.X\u2080\u2081 # T\u2081] - d[p.X\u2080\u2081 # X\u2081] + I[T\u2081 : T\u2083] / 2) := by\n have : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), sum2]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n gcongr\n linarith [condRuzsaDist_le' \u2119 \u2119 p.hmeas1 hT\u2081 hT\u2083]\n\n have h3 : p.\u03b7 * sum3 \u2264 p.\u03b7 * (d[p.X\u2080\u2082 # T\u2082] - d[p.X\u2080\u2082 # X\u2082] + I[T\u2082 : T\u2083] / 2) := by\n have : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp [sum3, integral_sub (.of_finite _ _) (.of_finite _ _)]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n gcongr\n linarith [condRuzsaDist_le' \u2119 \u2119 p.hmeas2 hT\u2082 hT\u2083]\n\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * c[T\u2081 # T\u2082] + p.\u03b7 * (I[T\u2081 : T\u2083] + I[T\u2082 : T\u2083]) / 2 := by\n suffices sum4 = sum1 + p.\u03b7 * (sum2 + sum3) by linarith\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _), integral_mul_left]\n\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $ ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\n/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n-\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\n\nThen there exist random variables $T'_1, T'_2$ such that\n\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X _1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2])$$\n\nis at most\n\n$$\\delta + \\frac{\\eta}{3} \\biggl( \\delta + \\sum_{i=1}^2 \\sum_{j = 1}^3\n (d[X^0_i;T_j] - d[X^0_i; X_i]) \\biggr).$$\n-/\nlemma construct_good : k \u2264 \u03b4 + (p.\u03b7/3) * (\u03b4 + c[T\u2081 # T\u2081] + c[T\u2082 # T\u2082] + c[T\u2083 # T\u2083]) := by\n have v2 := construct_good_prelim p X\u2081 X\u2082 h_min (by rw [\u2190 hT]; abel) hT\u2081 hT\u2083 hT\u2082\n have v3 := construct_good_prelim p X\u2081 X\u2082 h_min (by rw [\u2190 hT]; abel) hT\u2082 hT\u2081 hT\u2083\n have v6 := construct_good_prelim p X\u2081 X\u2082 h_min (by rw [\u2190 hT]; abel) hT\u2083 hT\u2082 hT\u2081\n simp only [mutualInfo, entropy_comm hT\u2082 hT\u2081, entropy_comm hT\u2083 hT\u2081, entropy_comm hT\u2083 hT\u2082]\n at *\n linarith\n\nlemma construct_good' (\u03bc : Measure \u03a9') [IsProbabilityMeasure \u03bc]:\n k \u2264 \u03b4[\u03bc] + (p.\u03b7/3) * (\u03b4[\u03bc] + c[T\u2081 ; \u03bc # T\u2081 ; \u03bc] + c[T\u2082 ; \u03bc # T\u2082 ; \u03bc] + c[T\u2083 ; \u03bc # T\u2083 ; \u03bc]) := by\n letI : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n apply construct_good p X\u2081 X\u2082 h_min hT hT\u2081 hT\u2082 hT\u2083\n\nlemma cond_c_eq_integral {Y Z : \u03a9' \u2192 G} (hY : Measurable Y) (hZ : Measurable Z) : c[Y | Z # Y | Z] =\n (Measure.map Z \u2119)[fun z => c[Y ; \u2119[|Z \u2190 z] # Y ; \u2119[|Z \u2190 z]]] := by\n simp only [integral_eq_sum, smul_sub, smul_add, smul_sub, Finset.sum_sub_distrib,\n Finset.sum_add_distrib]\n simp_rw [\u2190 integral_eq_sum]\n rw [\u2190 condRuzsaDist'_eq_integral _ hY hZ, \u2190 condRuzsaDist'_eq_integral _ hY hZ, integral_const,\n integral_const]\n have : IsProbabilityMeasure (Measure.map Z \u2119) := isProbabilityMeasure_map hZ.aemeasurable\n simp only [measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul]\n\nvariable {R : \u03a9' \u2192 G} (hR : Measurable R)\nlocal notation3:max \"\u03b4'\" => I[T\u2081 : T\u2082|R] + I[T\u2082 : T\u2083|R] + I[T\u2083 : T\u2081|R]\n\nlemma delta'_eq_integral : \u03b4' = (Measure.map R \u2119)[fun r => \u03b4[\u2119[|R\u207b\u00b9' {r}]]] := by\n simp_rw [condMutualInfo_eq_integral_mutualInfo, integral_eq_sum, smul_add,\n Finset.sum_add_distrib]\n\nlemma cond_construct_good :\n k \u2264 \u03b4' + (p.\u03b7/3) * (\u03b4' + c[T\u2081 | R # T\u2081 | R] + c[T\u2082 | R # T\u2082 | R] + c[T\u2083 | R # T\u2083 | R]) := by\n rw [delta'_eq_integral, cond_c_eq_integral _ _ _ hT\u2081 hR, cond_c_eq_integral _ _ _ hT\u2082 hR,\n cond_c_eq_integral _ _ _ hT\u2083 hR]\n simp_rw [integral_eq_sum, \u2190 Finset.sum_add_distrib, \u2190 smul_add, Finset.mul_sum, mul_smul_comm,\n \u2190 Finset.sum_add_distrib, \u2190 smul_add]\n simp_rw [\u2190 integral_eq_sum]\n have : IsProbabilityMeasure (Measure.map R \u2119) := isProbabilityMeasure_map (by measurability)\n calc\n k = (Measure.map R \u2119)[fun _r => k] := by\n rw [integral_const]; simp\n _ \u2264 _ := ?_\n simp_rw [integral_eq_sum]\n apply Finset.sum_le_sum\n intro r _\n by_cases hr : \u2119 (R\u207b\u00b9' {r}) = 0\n \u00b7 rw [Measure.map_apply hR (MeasurableSet.singleton r), hr]\n simp\n simp_rw [smul_eq_mul]\n gcongr (?_ * ?_)\n \u00b7 apply rdist_nonneg hX\u2081 hX\u2082\n \u00b7 rfl\n have : IsProbabilityMeasure (\u2119[|R \u207b\u00b9' {r}]) := cond_isProbabilityMeasure \u2119 hr\n apply construct_good' p X\u2081 X\u2082 h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend construct_good\n\n", "theoremStatement": "/-- If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that\nPhrased in the contrapositive form for convenience of proof. -/\ntheorem tau_strictly_decreases_aux (hp\u03b7: p.\u03b7 = 1/9): d[X\u2081 # X\u2082] = 0 ", "theoremName": "tau_strictly_decreases_aux", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "fafcc9e942a4844a347223ca0e5ade8d7f76bfb5", "date": "2023-11-27"}, "file": "PFR/PFR/Endgame.lean", 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"Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h0 := cond_construct_good p X\u2081 X\u2082 hX\u2081 hX\u2082 h_min (sum_uvw_eq_zero ..)\n (show Measurable U by measurability) (show Measurable V by measurability)\n (show Measurable W by measurability) (show Measurable S by measurability)\n have h1 := sum_condMutual_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n have h2 := sum_dist_diff_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n have h_indep' : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'] := by\n let \u03c3 : Fin 4 \u2243 Fin 4 :=\n { toFun := ![0, 1, 3, 2]\n invFun := ![0, 1, 3, 2]\n left_inv := by intro i; fin_cases i <;> rfl\n right_inv := by intro i; fin_cases i <;> rfl }\n refine' iIndepFun.reindex \u03c3.symm _\n convert h_indep using 1\n ext i; fin_cases i <;> rfl\n have h3 := first_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep' h_min\n have h : k \u2264 (8*p.\u03b7 + p.\u03b7^2) * k := calc\n k \u2264 (1+p.\u03b7/3) * (6*p.\u03b7*k - (1-5*p.\u03b7) / (1-p.\u03b7) * (2*p.\u03b7*k - I\u2081)) + p.\u03b7/3*((6-3*p.\u03b7)*k + 3*(2*p.\u03b7*k-I\u2081)) := by\n rw [hp\u03b7] at *\n linarith\n _ = (8*p.\u03b7+p.\u03b7^2)*k - ((1-5*p.\u03b7)/(1-p.\u03b7)*(1+p.\u03b7/3)-p.\u03b7)*(2*p.\u03b7*k-I\u2081) := by\n ring\n _ \u2264 (8*p.\u03b7 + p.\u03b7^2) * k := by\n rw [hp\u03b7] at *\n norm_num\n linarith\n have : 0 \u2264 k := rdist_nonneg hX\u2081 hX\u2082\n rw [hp\u03b7] at *\n linarith", "proofType": "tactic", "proofLengthLines": 28, "proofLengthTokens": 1279}} +{"srcContext": "import PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic\nimport PFR.Mathlib.Probability.Kernel.Composition\nimport PFR.ForMathlib.Entropy.Kernel.Basic\n\n/-!\n# Mutual Information of kernels\n\n## Main definitions\n\n* `mutualInfo`: Mutual information of a kernel `\u03ba` into a product space with respect to a\n measure `\u03bc`. This is denoted by `Ik[\u03ba, \u03bc]` and is equal to\n `Hk[fst \u03ba, \u03bc] + Hk[snd \u03ba, \u03bc] - Hk[\u03ba, \u03bc]`.\n\n## Main statements\n\n* `mutualInfo_nonneg`: `Ik[\u03ba, \u03bc]` is nonnegative\n* `entropy_condKernel_le_entropy_fst` and `entropy_condKernel_le_entropy_snd`: conditioning\n reduces entropy.\n\n## Notations\n\n* `Ik[\u03ba, \u03bc] = kernel.entropy \u03ba \u03bc`\n\n-/\n\nopen Function MeasureTheory Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory.kernel\n\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S]\n [Nonempty T] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U]\n [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V]\n {\u03ba : kernel T S} {\u03bc : Measure T} {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 U}\n\n/-- Mutual information of a kernel into a product space with respect to a measure. -/\nnoncomputable\ndef mutualInfo (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) : \u211d :=\n Hk[fst \u03ba, \u03bc] + Hk[snd \u03ba, \u03bc] - Hk[\u03ba, \u03bc]\n\n/-- Mutual information of a kernel into a product space with respect to a measure. -/\nnotation3:100 \"Ik[\" \u03ba \" , \" \u03bc \"]\" => kernel.mutualInfo \u03ba \u03bc\n\nlemma mutualInfo_def (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) :\n Ik[\u03ba, \u03bc] = Hk[fst \u03ba, \u03bc] + Hk[snd \u03ba, \u03bc] - Hk[\u03ba, \u03bc] := rfl\n\n@[simp]\nlemma mutualInfo_zero_measure (\u03ba : kernel T (S \u00d7 U)) : Ik[\u03ba, (0 : Measure T)] = 0 := by\n simp [mutualInfo]\n\n@[simp]\nlemma mutualInfo_zero_kernel (\u03bc : Measure T) : Ik[(0 : kernel T (S \u00d7 U)), \u03bc] = 0 := by\n simp [mutualInfo]\n\nlemma mutualInfo_congr {\u03ba \u03b7 : kernel T (S \u00d7 U)} {\u03bc : Measure T} (h : \u03ba =\u1d50[\u03bc] \u03b7) :\n Ik[\u03ba, \u03bc] = Ik[\u03b7, \u03bc] := by\n rw [mutualInfo, mutualInfo]\n have h1 : fst \u03ba =\u1d50[\u03bc] fst \u03b7 := by\n filter_upwards [h] with t ht\n rw [fst_apply, ht, fst_apply]\n have h2 : snd \u03ba =\u1d50[\u03bc] snd \u03b7 := by\n filter_upwards [h] with t ht\n rw [snd_apply, ht, snd_apply]\n rw [entropy_congr h1, entropy_congr h2, entropy_congr h]\n\nlemma mutualInfo_compProd {\u03ba : kernel T S} [IsMarkovKernel \u03ba]\n {\u03b7 : kernel (T \u00d7 S) U} [IsMarkovKernel \u03b7] {\u03bc : Measure T} [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7 : AEFiniteKernelSupport \u03b7 (\u03bc \u2297\u2098 \u03ba)):\n Ik[\u03ba \u2297\u2096 \u03b7, \u03bc] = Hk[\u03ba, \u03bc] + Hk[snd (\u03ba \u2297\u2096 \u03b7), \u03bc] - Hk[\u03ba \u2297\u2096 \u03b7, \u03bc] := by\n rw [mutualInfo, entropy_compProd h\u03ba h\u03b7, fst_compProd]\n\nlemma mutualInfo_eq_snd_sub {\u03ba : kernel T (S \u00d7 U)} [IsMarkovKernel \u03ba]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba: AEFiniteKernelSupport \u03ba \u03bc) :\n Ik[\u03ba, \u03bc] = Hk[snd \u03ba, \u03bc] - Hk[condKernel \u03ba, \u03bc \u2297\u2098 (fst \u03ba)] := by\n rw [mutualInfo, chain_rule h\u03ba]\n ring\n\nlemma mutualInfo_eq_fst_sub {\u03ba : kernel T (S \u00d7 U)} [IsMarkovKernel \u03ba]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba: AEFiniteKernelSupport \u03ba \u03bc) :\n Ik[\u03ba, \u03bc] = Hk[fst \u03ba, \u03bc] - Hk[condKernel (swapRight \u03ba), \u03bc \u2297\u2098 (snd \u03ba)] := by\n rw [mutualInfo, chain_rule' h\u03ba]\n ring\n\n@[simp]\nlemma mutualInfo_prod {\u03ba : kernel T S} {\u03b7 : kernel T U} [IsMarkovKernel \u03ba] [IsMarkovKernel \u03b7]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba: AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7: AEFiniteKernelSupport \u03b7 \u03bc) :\n Ik[\u03ba \u00d7\u2096 \u03b7, \u03bc] = 0 := by\n rw [mutualInfo, snd_prod, fst_prod, entropy_prod h\u03ba h\u03b7, sub_self]\n\n@[simp]\nlemma mutualInfo_swapRight (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) :\n Ik[swapRight \u03ba, \u03bc] = Ik[\u03ba, \u03bc] := by\n rw [mutualInfo, fst_swapRight, snd_swapRight, entropy_swapRight, add_comm]\n rfl\n\nlemma mutualInfo_nonneg' {\u03ba : kernel T (S \u00d7 U)} {\u03bc : Measure T} [IsFiniteMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : FiniteKernelSupport \u03ba) :\n 0 \u2264 Ik[\u03ba, \u03bc] := by\n simp_rw [mutualInfo, entropy, integral_eq_sum' _ (measure_compl_support \u03bc),\n smul_eq_mul]\n rw [\u2190 Finset.sum_add_distrib, \u2190 Finset.sum_sub_distrib]\n refine Finset.sum_nonneg (fun x _ \u21a6 ?_)\n by_cases hx : \u03bc {x} = 0\n \u00b7 simp [hx]\n rw [\u2190 mul_add, \u2190 mul_sub]\n refine mul_nonneg ENNReal.toReal_nonneg ?_\n rw [fst_apply, snd_apply]\n have : FiniteSupport (\u03ba x) := \u27e8h\u03ba x\u27e9\n exact measureMutualInfo_nonneg\n\nlemma mutualInfo_nonneg {\u03ba : kernel T (S \u00d7 U)} {\u03bc : Measure T} [IsFiniteMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n 0 \u2264 Ik[\u03ba, \u03bc] := by\n rw [mutualInfo_congr h\u03ba.ae_eq_mk]\n exact mutualInfo_nonneg' h\u03ba.finiteKernelSupport_mk\n\nlemma entropy_condKernel_le_entropy_fst (\u03ba : kernel T (S \u00d7 U)) [IsMarkovKernel \u03ba]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[condKernel (swapRight \u03ba), \u03bc \u2297\u2098 (snd \u03ba)] \u2264 Hk[fst \u03ba, \u03bc] := by\n rw [\u2190 sub_nonneg, \u2190 mutualInfo_eq_fst_sub h\u03ba]\n exact mutualInfo_nonneg h\u03ba\n\nlemma entropy_condKernel_le_entropy_snd {\u03ba : kernel T (S \u00d7 U)} [IsMarkovKernel \u03ba]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[condKernel \u03ba, \u03bc \u2297\u2098 (fst \u03ba)] \u2264 Hk[snd \u03ba, \u03bc] := by\n rw [\u2190 sub_nonneg, \u2190 mutualInfo_eq_snd_sub h\u03ba]\n exact mutualInfo_nonneg h\u03ba\n\n-- TODO: extract lemma(s) from this:\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_of_injective {V : Type*} [Nonempty V] [Countable V]\n [MeasurableSpace V] [MeasurableSingletonClass V]\n (\u03ba : kernel T (S \u00d7 U)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n (f : S \u00d7 U \u2192 V) (hfi : \u2200 x, Injective (fun y \u21a6 f (x, y))) [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba f (measurable_of_countable f), \u03bc] := by\n rw [mutualInfo_eq_snd_sub h\u03ba]\n have hf : Measurable f := measurable_of_countable f\n ring_nf\n calc\n Hk[condKernel \u03ba, \u03bc \u2297\u2098 fst \u03ba] = Hk[snd ((condKernel \u03ba) \u2297\u2096 (deterministic (fun x : (T \u00d7 S) \u00d7 U \u21a6 f (x.1.2, x.2)) (measurable_of_countable _))), \u03bc \u2297\u2098 fst \u03ba] := by\n symm\n apply entropy_snd_compProd_deterministic_of_injective\n simpa\n _ = Hk[condKernel (map \u03ba (fun p \u21a6 (p.1, f p)) (measurable_fst.prod_mk hf)),\n \u03bc \u2297\u2098 fst \u03ba] := entropy_congr (condKernel_map_prod_mk_left \u03ba \u03bc f).symm\n _ = Hk[condKernel (map \u03ba (fun p \u21a6 (p.1, f p)) (measurable_fst.prod_mk hf)),\n \u03bc \u2297\u2098 fst (map \u03ba (fun p \u21a6 (p.1, f p)) (measurable_fst.prod_mk hf))] := by\n congr 2 with x\n rw [fst_map_prod _ measurable_fst hf, fst_apply, map_apply]\n _ \u2264 Hk[snd (map \u03ba (fun p \u21a6 (p.1, f p)) (measurable_fst.prod_mk hf)), \u03bc] :=\n entropy_condKernel_le_entropy_snd (h\u03ba.map _)\n _ = Hk[map \u03ba f hf, \u03bc] := by rw [snd_map_prod _ measurable_fst]\n\nlemma compProd_assoc (\u03be : kernel T S) [IsMarkovKernel \u03be]\n (\u03ba : kernel (T \u00d7 S) U) [IsMarkovKernel \u03ba] (\u03b7 : kernel (T \u00d7 S \u00d7 U) V) [IsMarkovKernel \u03b7] :\n map ((\u03be \u2297\u2096 \u03ba) \u2297\u2096 \u03b7) MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable\n = \u03be \u2297\u2096 (\u03ba \u2297\u2096 (comap \u03b7 MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable)) := by\n ext x s hs\n rw [map_apply' _ _ _ hs, compProd_apply _ _ _ (MeasurableEquiv.prodAssoc.measurable hs),\n compProd_apply _ _ _ hs, lintegral_compProd]\n swap; \u00b7 exact measurable_kernel_prod_mk_left' (MeasurableEquiv.prodAssoc.measurable hs) _\n congr with a\n rw [compProd_apply]\n swap; \u00b7 exact measurable_prod_mk_left hs\n congr\n\nlemma Measure.compProd_compProd (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n (\u03be : kernel T S) [IsMarkovKernel \u03be] (\u03ba : kernel (T \u00d7 S) U) [IsMarkovKernel \u03ba] :\n \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba) = (\u03bc \u2297\u2098 \u03be \u2297\u2098 \u03ba).map MeasurableEquiv.prodAssoc := by\n ext s hs\n rw [Measure.compProd_apply hs, Measure.map_apply MeasurableEquiv.prodAssoc.measurable hs,\n Measure.compProd_apply (MeasurableEquiv.prodAssoc.measurable hs),\n Measure.lintegral_compProd]\n swap; \u00b7 exact measurable_kernel_prod_mk_left (MeasurableEquiv.prodAssoc.measurable hs)\n congr with a\n rw [compProd_apply _ _ _ (measurable_prod_mk_left hs)]\n congr\n\nlemma Measure.compProd_compProd' (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n (\u03be : kernel T S) [IsMarkovKernel \u03be] (\u03ba : kernel (T \u00d7 S) U) [IsMarkovKernel \u03ba] :\n \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba) = (\u03bc \u2297\u2098 \u03be \u2297\u2098 \u03ba).comap\n (MeasurableEquiv.prodAssoc.symm : T \u00d7 S \u00d7 U \u2243\u1d50 (T \u00d7 S) \u00d7 U) := by\n rw [MeasurableEquiv.comap_symm, Measure.compProd_compProd]\n\nlemma Measure.compProd_compProd'' (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n (\u03be : kernel T S) [IsMarkovKernel \u03be] (\u03ba : kernel (T \u00d7 S) U) [IsMarkovKernel \u03ba] :\n \u03bc \u2297\u2098 \u03be \u2297\u2098 \u03ba = Measure.comap MeasurableEquiv.prodAssoc (\u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)) := by\n rw [Measure.compProd_compProd, \u2190 MeasurableEquiv.map_symm, Measure.map_map]\n \u00b7 simp\n \u00b7 exact MeasurableEquiv.prodAssoc.symm.measurable\n \u00b7 exact MeasurableEquiv.prodAssoc.measurable\n\n-- from kernel (T \u00d7 S \u00d7 U) V ; Measure (T \u00d7 S \u00d7 U)\n-- to kernel (T \u00d7 S) V ; Measure (T \u00d7 S)\nlemma entropy_submodular_compProd {\u03be : kernel T S} [IsMarkovKernel \u03be]\n {\u03ba : kernel (T \u00d7 S) U} [IsMarkovKernel \u03ba] {\u03b7 : kernel (T \u00d7 S \u00d7 U) V} [IsMarkovKernel \u03b7]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba (\u03bc \u2297\u2098 \u03be))\n (h\u03b7 : AEFiniteKernelSupport \u03b7 (\u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba))) (h\u03be : AEFiniteKernelSupport \u03be \u03bc) :\n Hk[\u03b7, \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)]\n \u2264 Hk[snd (\u03ba \u2297\u2096 (comap \u03b7 MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable)), \u03bc \u2297\u2098 \u03be] := by\n have h_meas := (MeasurableEquiv.prodAssoc.symm : T \u00d7 S \u00d7 U \u2243\u1d50 (T \u00d7 S) \u00d7 U).symm.measurable\n have : FiniteSupport (\u03bc \u2297\u2098 \u03be) := finiteSupport_of_compProd h\u03be\n have : FiniteSupport (\u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)) := finiteSupport_of_compProd (h\u03be.compProd h\u03ba)\n have h := entropy_condKernel_le_entropy_snd\n (\u03ba := \u03ba \u2297\u2096 (comap \u03b7 MeasurableEquiv.prodAssoc h_meas)) (\u03bc := \u03bc \u2297\u2098 \u03be) ?_\n simp only [fst_compProd] at h\n have : condKernel (\u03ba \u2297\u2096 comap \u03b7 \u2191MeasurableEquiv.prodAssoc h_meas)\n =\u1d50[\u03bc \u2297\u2098 \u03be \u2297\u2098 \u03ba] comap \u03b7 \u2191MeasurableEquiv.prodAssoc h_meas := by\n exact condKernel_compProd_ae_eq \u03ba (comap \u03b7 \u2191MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable)\n (\u03bc \u2297\u2098 \u03be)\n rw [entropy_congr this, Measure.compProd_compProd'', entropy_comap_equiv] at h\n . exact h\n \u00b7 refine (h\u03ba.compProd ?_)\n convert h\u03b7.comap_equiv MeasurableEquiv.prodAssoc\n exact Measure.compProd_compProd'' _ _ _\n\nlemma entropy_condKernel_compProd_triple (\u03be : kernel T S) [IsMarkovKernel \u03be]\n (\u03ba : kernel (T \u00d7 S) U) [IsMarkovKernel \u03ba] (\u03b7 : kernel (T \u00d7 S \u00d7 U) V) [IsMarkovKernel \u03b7]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] :\n Hk[condKernel (\u03be \u2297\u2096 \u03ba \u2297\u2096 \u03b7) , \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)] = Hk[\u03b7, \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)] :=\n entropy_congr (condKernel_compProd_ae_eq (\u03be \u2297\u2096 \u03ba) \u03b7 \u03bc)\n\n/- $$ H[X,Y,Z] + H[X] \\leq H[Z,X] + H[Y,X].$$ -/\nlemma entropy_compProd_triple_add_entropy_le {\u03be : kernel T S} [IsMarkovKernel \u03be]\n {\u03ba : kernel (T \u00d7 S) U} [IsMarkovKernel \u03ba] {\u03b7 : kernel (T \u00d7 S \u00d7 U) V} [IsMarkovKernel \u03b7]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba (\u03bc \u2297\u2098 \u03be))\n (h\u03b7 : AEFiniteKernelSupport \u03b7 (\u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba))) (h\u03be : AEFiniteKernelSupport \u03be \u03bc) :\n Hk[(\u03be \u2297\u2096 \u03ba) \u2297\u2096 \u03b7, \u03bc] + Hk[\u03be, \u03bc]\n \u2264 Hk[\u03be \u2297\u2096 snd (\u03ba \u2297\u2096 comap \u03b7 MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable), \u03bc]\n + Hk[\u03be \u2297\u2096 \u03ba, \u03bc] := by\n rw [chain_rule,\n chain_rule (\u03ba := \u03be \u2297\u2096 snd (\u03ba \u2297\u2096 comap \u03b7 \u2191MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable))]\n simp only [fst_compProd, entropy_condKernel_compProd_triple, fst_deleteMiddle]\n . calc Hk[\u03be \u2297\u2096 \u03ba , \u03bc] + Hk[\u03b7 , \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)] + Hk[\u03be , \u03bc]\n = Hk[\u03be , \u03bc] + Hk[\u03be \u2297\u2096 \u03ba , \u03bc] + Hk[\u03b7 , \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)] := by abel\n _ \u2264 Hk[\u03be , \u03bc] + Hk[\u03be \u2297\u2096 \u03ba , \u03bc]\n + Hk[condKernel (\u03be \u2297\u2096 snd (\u03ba \u2297\u2096 comap \u03b7 MeasurableEquiv.prodAssoc _)) , \u03bc \u2297\u2098 \u03be] := by\n refine add_le_add le_rfl ?_\n refine (entropy_submodular_compProd h\u03ba h\u03b7 h\u03be).trans_eq ?_\n refine entropy_congr ?_\n exact (condKernel_compProd_ae_eq _ _ _).symm\n _ = Hk[\u03be , \u03bc] + Hk[condKernel (\u03be \u2297\u2096 snd (\u03ba \u2297\u2096 comap \u03b7 MeasurableEquiv.prodAssoc _)) , \u03bc \u2297\u2098 \u03be] + Hk[\u03be \u2297\u2096 \u03ba , \u03bc] := by abel\n . refine h\u03be.compProd ?_\n refine AEFiniteKernelSupport.snd ?_\n refine h\u03ba.compProd ?_\n convert h\u03b7.comap_equiv MeasurableEquiv.prodAssoc\n exact Measure.compProd_compProd'' _ _ _\n \u00b7 exact (h\u03be.compProd h\u03ba).compProd h\u03b7\n\n", "theoremStatement": "/-- The submodularity inequality:\n$$ H[X,Y,Z] + H[X] \\leq H[X,Z] + H[X,Y].$$ -/\nlemma entropy_triple_add_entropy_le' {\u03ba : kernel T (S \u00d7 U \u00d7 V)} [IsMarkovKernel \u03ba]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[\u03ba, \u03bc] + Hk[fst \u03ba, \u03bc] \u2264 Hk[deleteMiddle \u03ba, \u03bc] + Hk[deleteRight \u03ba, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_triple_add_entropy_le'", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "1e419c3b00cc596de5e8a17b30b679144993db04", "date": "2023-11-21"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/MutualInfo.lean", 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"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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.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.Lemma", "Mathlib.Tactic.TypeStar", "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.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.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.Init.Data.Int.Basic", "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.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "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.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.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.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "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.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.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n set \u03ba' := map \u03ba MeasurableEquiv.prodAssoc.symm MeasurableEquiv.prodAssoc.symm.measurable\n with h\u03ba'_def\n let \u03be := fst (fst \u03ba')\n let \u03ba'' := condKernel (fst \u03ba')\n let \u03b7 := condKernel \u03ba'\n have h\u03be_eq : \u03be = fst \u03ba := by\n simp only [\u03be, \u03ba', fst, MeasurableEquiv.prodAssoc, MeasurableEquiv.coe_mk, Equiv.coe_fn_mk,\n map_map]\n congr\n have h_compProd_eq : \u03be \u2297\u2096 \u03ba'' = fst \u03ba' := (disintegration (fst \u03ba')).symm\n have h_compProd_triple_eq : (\u03be \u2297\u2096 \u03ba'') \u2297\u2096 \u03b7 = \u03ba' := by\n rw [h_compProd_eq]\n exact (disintegration \u03ba').symm\n have h_compProd_triple_eq' : \u03be \u2297\u2096 (\u03ba'' \u2297\u2096 comap \u03b7 MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable)\n = \u03ba := by\n rw [\u2190 compProd_assoc, h_compProd_triple_eq,h\u03ba'_def, map_map]\n simp\n have h := entropy_compProd_triple_add_entropy_le (\u03be := \u03be) (\u03ba := \u03ba'') (\u03b7 := \u03b7) (\u03bc := \u03bc) ?_ ?_ ?_\n rotate_left\n . exact aefiniteKernelSupport_of_cond _ (h\u03ba.map _).fst\n . rw [h_compProd_eq]\n apply aefiniteKernelSupport_of_cond\n exact h\u03ba.map _\n \u00b7 exact (h\u03ba.map _).fst.fst\n rw [\u2190 h\u03be_eq]\n have h_right : deleteRight \u03ba = fst \u03ba' := by\n simp only [\u03ba', deleteRight, fst, map_map]\n congr\n have h_middle : deleteMiddle \u03ba\n = \u03be \u2297\u2096 snd (\u03ba'' \u2297\u2096 comap \u03b7 MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable) := by\n rw [\u2190 deleteMiddle_compProd, h_compProd_triple_eq']\n have h\u03ba : Hk[\u03ba, \u03bc] = Hk[\u03ba', \u03bc] := by\n rw [h\u03ba'_def, entropy_map_of_injective]\n exact MeasurableEquiv.prodAssoc.symm.injective\n rw [h_right, h_middle, h\u03ba, \u2190 h_compProd_triple_eq, fst_compProd]\n . exact h", "proofType": "tactic", "proofLengthLines": 36, "proofLengthTokens": 1543}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] := by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]\n\n/-- Adding a constant to a random variable does not change the Rusza distance. -/\nlemma rdist_add_const [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y + fun _ \u21a6 c; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n have A : IdentDistrib (Y' + fun _ \u21a6 c) (Y + fun _ \u21a6 c) \u03bd \u03bc' := by\n change IdentDistrib (fun \u03c9 \u21a6 Y' \u03c9 + c) (fun \u03c9 \u21a6 Y \u03c9 + c) \u03bd \u03bc'\n apply hIdY.comp (measurable_add_const c)\n have B : IndepFun X' (Y' + fun _ \u21a6 c) \u03bd := by\n change IndepFun X' (fun \u03c9 \u21a6 Y' \u03c9 + c) \u03bd\n apply hind.comp measurable_id (measurable_add_const c)\n have C : X' - (Y' + fun _ \u21a6 c) = (X' - Y') + (fun _ \u21a6 -c) := by\n ext \u03c9; simp; abel\n rw [\u2190 hIdX.rdist_eq hIdY, \u2190 hIdX.rdist_eq A, hind.rdist_eq hX' hY',\n B.rdist_eq hX' (hY'.add_const _), entropy_add_const hY' c, C, entropy_add_const]\n exact hX'.sub hY'\n\n/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\nlemma rdist_add_const' [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] (c: G) (c': G)\n (hX : Measurable X) (hY : Measurable Y) :\n d[X + fun _ \u21a6 c; \u03bc # Y + fun _ \u21a6 c'; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_add_const _ hY, rdist_symm, rdist_add_const hY hX, rdist_symm]\n measurability\n\n/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (X : \u03a9 \u2192 G) (Y : \u03a9 \u2192 G) (Z : \u03a9 \u2192 G)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n have h1 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] + H[X - Y; \u03bc] \u2264 H[\u27e8X - Z, X - Y\u27e9; \u03bc] + H[\u27e8Y, X - Y\u27e9; \u03bc] :=\n entropy_triple_add_entropy_le \u03bc (hX.sub hZ) hY (hX.sub hY)\n have h2 : H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n calc H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n have : \u27e8X - Z, X - Y\u27e9 = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 \u27e8X - Z, Y - Z\u27e9 := by ext1; simp\n rw [this]\n apply entropy_comp_le \u03bc (by measurability)\n _ \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n have h : 0 \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] - H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n apply mutualInfo_nonneg (by measurability) (by measurability) \u03bc\n linarith\n have h3 : H[\u27e8Y, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X, Y\u27e9 ; \u03bc] := by\n have : \u27e8Y, X - Y\u27e9 = (fun p \u21a6 (p.2, p.1 - p.2)) \u2218 \u27e8X, Y\u27e9 := by ext1; simp\n rw [this]\n exact entropy_comp_le \u03bc (hX.prod_mk hY) _\n have h4 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] = H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] := by\n refine entropy_of_comp_eq_of_comp \u03bc ((hX.sub hZ).prod_mk (hY.prod_mk (hX.sub hY)))\n (hX.prod_mk (hY.prod_mk hZ))\n (fun p : G \u00d7 (G \u00d7 G) \u21a6 (p.2.2 + p.2.1, p.2.1, -p.1 + p.2.2 + p.2.1))\n (fun p : G \u00d7 G \u00d7 G \u21a6 (p.1 - p.2.2, p.2.1, p.1 - p.2.1)) ?_ ?_\n \u00b7 ext1; simp\n \u00b7 ext1; simp\n have h5 : H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [entropy_assoc hX hY hZ, entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h\n rw [h4, h5] at h1\n calc H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Y - Z; \u03bc] - H[Z; \u03bc] := by linarith\n _ = H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n congr 2\n rw [entropy_sub_comm hY hZ]\n\n/-- The **entropic Ruzsa triangle inequality** -/\nlemma rdist_triangle {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [h\u03bc : IsProbabilityMeasure \u03bc] [h\u03bc' : IsProbabilityMeasure \u03bc'] [h\u03bc'' : IsProbabilityMeasure \u03bc''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n d[X ; \u03bc # Z ; \u03bc''] \u2264 d[X ; \u03bc # Y ; \u03bc'] + d[Y ; \u03bc' # Z ; \u03bc''] := by\n obtain \u27e8A, mA, \u03bcA, X', Y', Z', h\u03bcA, hInd, hX', hY', hZ', HX, HY, HZ, _, _, _\u27e9 :=\n independent_copies3_nondep_finiteRange hX hY hZ \u03bc \u03bc' \u03bc''\n suffices d[X' ; \u03bcA # Z' ; \u03bcA] \u2264 d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] by\n rwa [HX.rdist_eq HY, HY.rdist_eq HZ, HX.rdist_eq HZ] at this\n have IndepLem : IndepFun (\u27e8X', Z'\u27e9) Y' \u03bcA :=\n iIndepFun.indepFun_prod_mk hInd (fun i => by fin_cases i ; all_goals { simpa }) 0 2 1\n (by norm_cast) (by norm_cast)\n calc d[X' ; \u03bcA # Z' ; \u03bcA] = H[X' - Z'; \u03bcA] - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) := by\n rw [ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using hInd.indepFun (show 0 \u2260 2 by norm_cast)) hX' hZ'] ; ring\n _ \u2264 (H[X' - Y' ; \u03bcA] + H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA]) - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) :=\n sub_le_sub_right (ent_of_diff_le _ _ _ hX' hZ' hY' IndepLem) _\n _ = (H[X' - Y' ; \u03bcA] - H[X'; \u03bcA] / 2 - H[Y' ; \u03bcA] / 2) +\n (H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA] / 2 - H[Z'; \u03bcA] / 2) := by ring\n _ = d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] := by\n rw [ProbabilityTheory.IndepFun.rdist_eq (by simpa using (iIndepFun.indepFun hInd\n (show 0 \u2260 1 by norm_cast))) hX' hY', ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using (iIndepFun.indepFun hInd (show 1 \u2260 2 by norm_cast))) hY' hZ']\n\nvariable [MeasurableSingletonClass S] [MeasurableSingletonClass T]\n\n/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) [IsFiniteMeasure \u03bc]\n (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc'\"]\" => condRuzsaDist X Z Y W \u03bc \u03bc'\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" # \" Y \" | \" W \"]\" => condRuzsaDist X Z Y W volume volume\n\nlemma condRuzsaDist_def (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W] := rfl\n\n/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\nlemma condRuzsaDist_symm {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hZ : Measurable Z)\n (hW : Measurable W) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange Z]\n [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[Y | W ; \u03bc' # X | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist_symm]\n\n@[simp] lemma condRuszaDist_zero_right (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n d[X | Z ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero,\n kernel.rdist_zero_right]\n\n@[simp] lemma condRuszaDist_zero_left (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; 0 # Y | W ; \u03bc'] = 0 := by\n simp [condRuzsaDist]\n\nlemma condRuzsaDist_nonneg {X : \u03a9 \u2192 G} (hX : Measurable X) [FiniteRange X]\n {Z : \u03a9 \u2192 S} (hZ : Measurable Z) [FiniteRange Z]\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) [FiniteRange Y]\n {W : \u03a9' \u2192 T} (hW : Measurable W) [FiniteRange W]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n 0 \u2264 d[X | Z ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def]\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n refine kernel.rdist_nonneg ?_ ?_\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hX hZ\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hY hW\n\n/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\nlemma rdist_eq_rdistm : d[X ; \u03bc # Y ; \u03bc'] = kernel.rdistm (\u03bc.map X) (\u03bc'.map Y) := rfl\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\nlemma condRuzsaDist_eq_sum {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z in FiniteRange.toFinset Z, \u2211 w in FiniteRange.toFinset W,\n (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((((FiniteRange.toFinset Z)\n \u00d7\u02e2 (FiniteRange.toFinset W)) : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Finset.sum_product,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\nlemma condRuzsaDist_eq_sum' {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [Fintype S] [Fintype T] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z, \u2211 w, (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Fintype.sum_prod_type,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist' (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc' \"]\" => condRuzsaDist' X Y W \u03bc \u03bc'\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" # \" Y \" | \" W \"]\" => condRuzsaDist' X Y W volume volume\n\n/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\nlemma condRuzsaDist'_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y | W ; \u03bc'] =\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W] :=\n rfl\n\n@[simp] lemma condRuzsaDist'_zero_right (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) :\n d[X ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist'_def, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero, kernel.rdist_zero_right]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\nlemma condRuzsaDist'_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (dirac ()) (\u03bc'.map W) ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 FiniteRange.toFinset W : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, smul_eq_mul, Finset.sum_product]\n simp only [Finset.univ_unique, PUnit.default_eq_unit, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, Pi.one_apply, one_mul,\n Finset.sum_singleton]\n simp_rw [Measure.map_apply hW (measurableSet_singleton _)]\n congr with w\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff]\n refine Or.inr ?_\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm, condDistrib_apply hY hW _ _ hw]\n congr\n\n/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\nlemma condRuzsaDist'_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_eq_sum hY hW \u03bc \u03bc']\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\nopen scoped ENNReal\n\nlemma condRuzsaDist'_prod_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [FiniteRange W] [FiniteRange W']:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n have : d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc'] = \u2211 w in ((FiniteRange.toFinset W') \u00d7\u02e2 FiniteRange.toFinset W), (\u03bc' ((fun a => (W' a, W a)) \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; \u03bc'[|(fun a => (W' a, W a)) \u207b\u00b9' {w}]] := by\n rw [condRuzsaDist'_eq_sum hY (hW'.prod_mk hW)]\n apply Finset.sum_subset\n . intro (t, t')\n simp\n intro \u03c9 h1 h2\n exact \u27e8\u27e8\u03c9, h1\u27e9, \u27e8\u03c9, h2\u27e9\u27e9\n intro (t, t') _ ht\n simp at ht\n have : (fun a => (W' a, W a)) \u207b\u00b9' {(t, t')} = \u2205 := by\n ext \u03c9\n simp\n exact ht \u03c9\n simp [this]\n rw [this, Finset.sum_product_right]\n congr 1 with w\n rw [condRuzsaDist'_eq_sum hY hW', Finset.mul_sum]\n congr 1 with w'\n rw [\u2190 mul_assoc]\n have A : (fun a \u21a6 (W' a, W a)) \u207b\u00b9' {(w', w)} = W' \u207b\u00b9' {w'} \u2229 W\u207b\u00b9' {w} := by ext; simp\n congr 1\n \u00b7 simp only [ProbabilityTheory.cond, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n smul_eq_mul, ENNReal.toReal_mul, A, restrict_apply (hW' (MeasurableSet.singleton w'))]\n rcases eq_bot_or_bot_lt (\u03bc' (W \u207b\u00b9' {w})) with hw|hw\n \u00b7 have : \u03bc' (W' \u207b\u00b9' {w'} \u2229 W \u207b\u00b9' {w}) = 0 :=\n le_antisymm (le_trans (measure_mono (Set.inter_subset_right _ _)) hw.le) bot_le\n simp [hw, this]\n \u00b7 rw [\u2190 mul_assoc, \u2190 ENNReal.toReal_mul, ENNReal.mul_inv_cancel, ENNReal.one_toReal, one_mul]\n exacts [hw.ne', by finiteness]\n \u00b7 congr 1\n rw [A, cond_cond_eq_cond_inter' _ (hW (.singleton w)) (hW' (.singleton w')), Set.inter_comm]\n finiteness\n\n/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\nlemma condRuzsaDist'_prod_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [Fintype T]:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_prod_eq_sum \u03bc \u03bc' hY hW' hW]\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\nlemma condRuzsaDist'_eq_integral (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = (\u03bc'.map W)[fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])]] := by\n rw [condRuzsaDist'_eq_sum hY hW]\n simp_rw [\u2190 smul_eq_mul]\n have : (\u03bc'.map W) (FiniteRange.toFinset W : Set T)\u1d9c = 0 := by\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n convert symm $ integral_eq_sum' (\u03bc'.map W) this (fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])])\n rw [Measure.map_apply hW (MeasurableSet.singleton _)]\n\n/-- Conditioning by a constant does not affect Ruzsa distance. -/\nlemma condRuzsaDist_of_const {X : \u03a9 \u2192 G} (hX : Measurable X) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (c : S)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange W] :\n d[X|(fun _ \u21a6 c) ; \u03bc # Y | W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def, condRuzsaDist'_def, Measure.map_const,measure_univ,one_smul, kernel.rdist,\n kernel.rdist, integral_prod, integral_dirac, integral_prod,integral_dirac]\n dsimp; congr; ext x; congr\n rw [condDistrib_apply hX measurable_const]\n \u00b7 simp\n \u00b7 simp\n \u00b7 exact integrable_of_finiteSupport _\n \u00b7 exact integrable_of_finiteSupport _\n\n", "theoremStatement": "/-- If $(X,Z)$ and $(Y,W)$ are independent, then\n$$ d[X | Z ; Y | W] = H[X'- Y'|Z', W'] - H[X'|Z']/2 - H[Y'|W']/2$$\n-/\nlemma condRuzsaDist_of_indep\n {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun (\u27e8X, Z\u27e9) (\u27e8Y, W\u27e9) \u03bc) [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc] = H[X - Y | \u27e8Z, W\u27e9 ; \u03bc] - H[X | Z ; \u03bc]/2 - H[Y | W ; \u03bc]/2 ", "theoremName": "condRuzsaDist_of_indep", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc621d65f3d5b43306793549b7af0b5c922d3", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 680, "tokenPositionInFile": 34676, "theoremPositionInFile": 47}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 15, "numPremises": 108, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", 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"Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ (hZ.prod_mk hW),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hW]\n swap; \u00b7 exact hX.sub hY\n congr 2\n have hZW : IndepFun Z W \u03bc := by\n have h' := IndepFun.comp h measurable_snd measurable_snd\n exact h'\n have hZW_map : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map Z).prod (\u03bc.map W) :=\n (indepFun_iff_map_prod_eq_prod_map_map hZ.aemeasurable hW.aemeasurable).mp hZW\n rw [\u2190 hZW_map]\n refine kernel.entropy_congr ?_\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (hZ.prod_mk hW) _ _).symm\n refine (this.symm.trans ?_).symm\n suffices kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft S (condDistrib Y W \u03bc)\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc by\n filter_upwards [this] with x hx\n rw [kernel.map_apply, kernel.map_apply, hx]\n . exact (condDistrib_eq_prod_of_indepFun hX hZ hY hW \u03bc h).symm", "proofType": "tactic", "proofLengthLines": 23, "proofLengthTokens": 1241}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n", "theoremStatement": "/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 ", "theoremName": "gen_ineq_10", "fileCreated": {"commit": 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"Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", 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"Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.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.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.Paths", "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", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "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", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "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", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "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", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]", "proofType": "tactic", "proofLengthLines": 16, "proofLengthTokens": 1042}} +{"srcContext": "import PFR.FirstEstimate\nimport PFR.SecondEstimate\nimport PFR.ForMathlib.Elementary\n\n/-!\n# Endgame\n\nThe endgame on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ be independent random variables, with $X_1,\\tilde X_1$ copies\n of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $d[X_1;X_2] = k$\n* $U := X_1 + X_2$\n* $V := \\tilde X_1 + X_2$\n* $W := X_1 + \\tilde X_1$\n* $S := X_1 + X_2 + \\tilde X_1 + \\tilde X_2$.\n* $I_1 := I[U : V | S]$\n* $I_2 := I[U : W | S]$\n* $I_3 := I[V : W | S]$ (not explicitly defined in Lean)\n\n# Main results:\n\n* `sum_condMutual_le` : An upper bound on the total conditional mutual information $I_1+I_2+I_3$.\n* `sum_dist_diff_le`: A sum of the \"costs\" of $U$, $V$, $W$.\n* `construct_good`: A construction of two random variables with small Ruzsa distance between them\n given some random variables with control on total cost, as well as total mutual information.\n-/\n\nopen MeasureTheory ProbabilityTheory\nopen scoped BigOperators\n\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [m\u03a9 : MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2081', X\u2082'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[U : V | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `V = X\u2081' + X\u2082`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[U : V | S]\n\n/-- `I\u2082 := I[U : W | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `W = X\u2081' + X\u2081`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[U : W | S]\n\n--(Mantas) this times out in the proof below\nprivate lemma hmeas2 :\n Measurable fun p : Fin 4 \u2192 G => ((p 0 + p 1, p 0 + p 2), p 0 + p 1 + p 2 + p 3) := by\n apply Measurable.prod\n \u00b7 apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n\n", "theoremStatement": "/-- The quantity $I_3 = I[V:W|S]$ is equal to $I_2$. -/\nlemma I\u2083_eq : I[V : W | S] = I\u2082 ", "theoremName": "I\u2083_eq", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "086a96d386cd4145da293c11db68590805f79068", "date": "2023-11-19"}, "file": "PFR/PFR/Endgame.lean", "module": "PFR.Endgame", "jsonFile": "PFR.Endgame.jsonl", "positionMetadata": {"lineInFile": 91, "tokenPositionInFile": 2988, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 161, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", 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"Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h_indep2 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081, X\u2082'] := by\n exact h_indep.reindex_four_cbad\n have hident : IdentDistrib (fun a (i : Fin 4) => ![X\u2081, X\u2082, X\u2081', X\u2082'] i a)\n (fun a (j : Fin 4) => ![X\u2081', X\u2082, X\u2081, X\u2082'] j a) := by\n exact { aemeasurable_fst := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n aemeasurable_snd := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n map_eq := by\n rw [\u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081, X\u2082, X\u2081', X\u2082'])\n (Fin.cases hX\u2081 <| Fin.cases hX\u2082 <| Fin.cases hX\u2081' <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep,\n \u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081', X\u2082, X\u2081, X\u2082'])\n (Fin.cases hX\u2081' <| Fin.cases hX\u2082 <| Fin.cases hX\u2081 <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep2]\n congr\n ext i\n fin_cases i\n all_goals simp [h\u2081.map_eq] }\n have hmeas1 : Measurable (fun p : Fin 4 \u2192 G => (p 0 + p 1, p 0 + p 1 + p 2 + p 3)) := by\n simp_all only [Matrix.cons_val', Matrix.empty_val', Matrix.cons_val_fin_one]\n apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n have hUVS : IdentDistrib (prod U S) (prod V S) := by\n convert (IdentDistrib.comp hident hmeas1); simp; abel\n have hUVWS : IdentDistrib (prod (prod U W) S) (prod (prod V W) S) := by\n convert (IdentDistrib.comp hident hmeas2) <;> simp <;> abel\n have hU : Measurable U := Measurable.add hX\u2081 hX\u2082\n have hV : Measurable V := Measurable.add hX\u2081' hX\u2082\n have hW : Measurable W := Measurable.add hX\u2081' hX\u2081\n have hS : Measurable S := by measurability\n rw [condMutualInfo_eq hV hW hS, condMutualInfo_eq hU hW hS, chain_rule'' \u2119 hU hS,\n chain_rule'' \u2119 hV hS, chain_rule'' \u2119 hW hS, chain_rule'' \u2119 _ hS, chain_rule'' \u2119 _ hS,\n IdentDistrib.entropy_eq hUVS, IdentDistrib.entropy_eq hUVWS]\n \u00b7 exact Measurable.prod (by exact hU) (by exact hW)\n \u00b7 exact Measurable.prod (by exact hV) (by exact hW)", "proofType": "tactic", "proofLengthLines": 49, "proofLengthTokens": 2436}} +{"srcContext": "import PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic\nimport PFR.Mathlib.Probability.Kernel.Composition\nimport PFR.ForMathlib.Entropy.Kernel.Basic\n\n/-!\n# Mutual Information of kernels\n\n## Main definitions\n\n* `mutualInfo`: Mutual information of a kernel `\u03ba` into a product space with respect to a\n measure `\u03bc`. This is denoted by `Ik[\u03ba, \u03bc]` and is equal to\n `Hk[fst \u03ba, \u03bc] + Hk[snd \u03ba, \u03bc] - Hk[\u03ba, \u03bc]`.\n\n## Main statements\n\n* `mutualInfo_nonneg`: `Ik[\u03ba, \u03bc]` is nonnegative\n* `entropy_condKernel_le_entropy_fst` and `entropy_condKernel_le_entropy_snd`: conditioning\n reduces entropy.\n\n## Notations\n\n* `Ik[\u03ba, \u03bc] = kernel.entropy \u03ba \u03bc`\n\n-/\n\nopen Function MeasureTheory Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory.kernel\n\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S]\n [Nonempty T] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U]\n [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V]\n {\u03ba : kernel T S} {\u03bc : Measure T} {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 U}\n\n/-- Mutual information of a kernel into a product space with respect to a measure. -/\nnoncomputable\ndef mutualInfo (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) : \u211d :=\n Hk[fst \u03ba, \u03bc] + Hk[snd \u03ba, \u03bc] - Hk[\u03ba, \u03bc]\n\n/-- Mutual information of a kernel into a product space with respect to a measure. -/\nnotation3:100 \"Ik[\" \u03ba \" , \" \u03bc \"]\" => kernel.mutualInfo \u03ba \u03bc\n\nlemma mutualInfo_def (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) :\n Ik[\u03ba, \u03bc] = Hk[fst \u03ba, \u03bc] + Hk[snd \u03ba, \u03bc] - Hk[\u03ba, \u03bc] := rfl\n\n@[simp]\nlemma mutualInfo_zero_measure (\u03ba : kernel T (S \u00d7 U)) : Ik[\u03ba, (0 : Measure T)] = 0 := by\n simp [mutualInfo]\n\n@[simp]\nlemma mutualInfo_zero_kernel (\u03bc : Measure T) : Ik[(0 : kernel T (S \u00d7 U)), \u03bc] = 0 := by\n simp [mutualInfo]\n\nlemma mutualInfo_congr {\u03ba \u03b7 : kernel T (S \u00d7 U)} {\u03bc : Measure T} (h : \u03ba =\u1d50[\u03bc] \u03b7) :\n Ik[\u03ba, \u03bc] = Ik[\u03b7, \u03bc] := by\n rw [mutualInfo, mutualInfo]\n have h1 : fst \u03ba =\u1d50[\u03bc] fst \u03b7 := by\n filter_upwards [h] with t ht\n rw [fst_apply, ht, fst_apply]\n have h2 : snd \u03ba =\u1d50[\u03bc] snd \u03b7 := by\n filter_upwards [h] with t ht\n rw [snd_apply, ht, snd_apply]\n rw [entropy_congr h1, entropy_congr h2, entropy_congr h]\n\nlemma mutualInfo_compProd {\u03ba : kernel T S} [IsMarkovKernel \u03ba]\n {\u03b7 : kernel (T \u00d7 S) U} [IsMarkovKernel \u03b7] {\u03bc : Measure T} [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7 : AEFiniteKernelSupport \u03b7 (\u03bc \u2297\u2098 \u03ba)):\n Ik[\u03ba \u2297\u2096 \u03b7, \u03bc] = Hk[\u03ba, \u03bc] + Hk[snd (\u03ba \u2297\u2096 \u03b7), \u03bc] - Hk[\u03ba \u2297\u2096 \u03b7, \u03bc] := by\n rw [mutualInfo, entropy_compProd h\u03ba h\u03b7, fst_compProd]\n\nlemma mutualInfo_eq_snd_sub {\u03ba : kernel T (S \u00d7 U)} [IsMarkovKernel \u03ba]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba: AEFiniteKernelSupport \u03ba \u03bc) :\n Ik[\u03ba, \u03bc] = Hk[snd \u03ba, \u03bc] - Hk[condKernel \u03ba, \u03bc \u2297\u2098 (fst \u03ba)] := by\n rw [mutualInfo, chain_rule h\u03ba]\n ring\n\nlemma mutualInfo_eq_fst_sub {\u03ba : kernel T (S \u00d7 U)} [IsMarkovKernel \u03ba]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba: AEFiniteKernelSupport \u03ba \u03bc) :\n Ik[\u03ba, \u03bc] = Hk[fst \u03ba, \u03bc] - Hk[condKernel (swapRight \u03ba), \u03bc \u2297\u2098 (snd \u03ba)] := by\n rw [mutualInfo, chain_rule' h\u03ba]\n ring\n\n@[simp]\nlemma mutualInfo_prod {\u03ba : kernel T S} {\u03b7 : kernel T U} [IsMarkovKernel \u03ba] [IsMarkovKernel \u03b7]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba: AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7: AEFiniteKernelSupport \u03b7 \u03bc) :\n Ik[\u03ba \u00d7\u2096 \u03b7, \u03bc] = 0 := by\n rw [mutualInfo, snd_prod, fst_prod, entropy_prod h\u03ba h\u03b7, sub_self]\n\n@[simp]\nlemma mutualInfo_swapRight (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) :\n Ik[swapRight \u03ba, \u03bc] = Ik[\u03ba, \u03bc] := by\n rw [mutualInfo, fst_swapRight, snd_swapRight, entropy_swapRight, add_comm]\n rfl\n\nlemma mutualInfo_nonneg' {\u03ba : kernel T (S \u00d7 U)} {\u03bc : Measure T} [IsFiniteMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : FiniteKernelSupport \u03ba) :\n 0 \u2264 Ik[\u03ba, \u03bc] := by\n simp_rw [mutualInfo, entropy, integral_eq_sum' _ (measure_compl_support \u03bc),\n smul_eq_mul]\n rw [\u2190 Finset.sum_add_distrib, \u2190 Finset.sum_sub_distrib]\n refine Finset.sum_nonneg (fun x _ \u21a6 ?_)\n by_cases hx : \u03bc {x} = 0\n \u00b7 simp [hx]\n rw [\u2190 mul_add, \u2190 mul_sub]\n refine mul_nonneg ENNReal.toReal_nonneg ?_\n rw [fst_apply, snd_apply]\n have : FiniteSupport (\u03ba x) := \u27e8h\u03ba x\u27e9\n exact measureMutualInfo_nonneg\n\nlemma mutualInfo_nonneg {\u03ba : kernel T (S \u00d7 U)} {\u03bc : Measure T} [IsFiniteMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n 0 \u2264 Ik[\u03ba, \u03bc] := by\n rw [mutualInfo_congr h\u03ba.ae_eq_mk]\n exact mutualInfo_nonneg' h\u03ba.finiteKernelSupport_mk\n\nlemma entropy_condKernel_le_entropy_fst (\u03ba : kernel T (S \u00d7 U)) [IsMarkovKernel \u03ba]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[condKernel (swapRight \u03ba), \u03bc \u2297\u2098 (snd \u03ba)] \u2264 Hk[fst \u03ba, \u03bc] := by\n rw [\u2190 sub_nonneg, \u2190 mutualInfo_eq_fst_sub h\u03ba]\n exact mutualInfo_nonneg h\u03ba\n\nlemma entropy_condKernel_le_entropy_snd {\u03ba : kernel T (S \u00d7 U)} [IsMarkovKernel \u03ba]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[condKernel \u03ba, \u03bc \u2297\u2098 (fst \u03ba)] \u2264 Hk[snd \u03ba, \u03bc] := by\n rw [\u2190 sub_nonneg, \u2190 mutualInfo_eq_snd_sub h\u03ba]\n exact mutualInfo_nonneg h\u03ba\n\n-- TODO: extract lemma(s) from this:\nlemma entropy_snd_sub_mutualInfo_le_entropy_map_of_injective {V : Type*} [Nonempty V] [Countable V]\n [MeasurableSpace V] [MeasurableSingletonClass V]\n (\u03ba : kernel T (S \u00d7 U)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n (f : S \u00d7 U \u2192 V) (hfi : \u2200 x, Injective (fun y \u21a6 f (x, y))) [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[snd \u03ba, \u03bc] - Ik[\u03ba, \u03bc] \u2264 Hk[map \u03ba f (measurable_of_countable f), \u03bc] := by\n rw [mutualInfo_eq_snd_sub h\u03ba]\n have hf : Measurable f := measurable_of_countable f\n ring_nf\n calc\n Hk[condKernel \u03ba, \u03bc \u2297\u2098 fst \u03ba] = Hk[snd ((condKernel \u03ba) \u2297\u2096 (deterministic (fun x : (T \u00d7 S) \u00d7 U \u21a6 f (x.1.2, x.2)) (measurable_of_countable _))), \u03bc \u2297\u2098 fst \u03ba] := by\n symm\n apply entropy_snd_compProd_deterministic_of_injective\n simpa\n _ = Hk[condKernel (map \u03ba (fun p \u21a6 (p.1, f p)) (measurable_fst.prod_mk hf)),\n \u03bc \u2297\u2098 fst \u03ba] := entropy_congr (condKernel_map_prod_mk_left \u03ba \u03bc f).symm\n _ = Hk[condKernel (map \u03ba (fun p \u21a6 (p.1, f p)) (measurable_fst.prod_mk hf)),\n \u03bc \u2297\u2098 fst (map \u03ba (fun p \u21a6 (p.1, f p)) (measurable_fst.prod_mk hf))] := by\n congr 2 with x\n rw [fst_map_prod _ measurable_fst hf, fst_apply, map_apply]\n _ \u2264 Hk[snd (map \u03ba (fun p \u21a6 (p.1, f p)) (measurable_fst.prod_mk hf)), \u03bc] :=\n entropy_condKernel_le_entropy_snd (h\u03ba.map _)\n _ = Hk[map \u03ba f hf, \u03bc] := by rw [snd_map_prod _ measurable_fst]\n\nlemma compProd_assoc (\u03be : kernel T S) [IsMarkovKernel \u03be]\n (\u03ba : kernel (T \u00d7 S) U) [IsMarkovKernel \u03ba] (\u03b7 : kernel (T \u00d7 S \u00d7 U) V) [IsMarkovKernel \u03b7] :\n map ((\u03be \u2297\u2096 \u03ba) \u2297\u2096 \u03b7) MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable\n = \u03be \u2297\u2096 (\u03ba \u2297\u2096 (comap \u03b7 MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable)) := by\n ext x s hs\n rw [map_apply' _ _ _ hs, compProd_apply _ _ _ (MeasurableEquiv.prodAssoc.measurable hs),\n compProd_apply _ _ _ hs, lintegral_compProd]\n swap; \u00b7 exact measurable_kernel_prod_mk_left' (MeasurableEquiv.prodAssoc.measurable hs) _\n congr with a\n rw [compProd_apply]\n swap; \u00b7 exact measurable_prod_mk_left hs\n congr\n\nlemma Measure.compProd_compProd (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n (\u03be : kernel T S) [IsMarkovKernel \u03be] (\u03ba : kernel (T \u00d7 S) U) [IsMarkovKernel \u03ba] :\n \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba) = (\u03bc \u2297\u2098 \u03be \u2297\u2098 \u03ba).map MeasurableEquiv.prodAssoc := by\n ext s hs\n rw [Measure.compProd_apply hs, Measure.map_apply MeasurableEquiv.prodAssoc.measurable hs,\n Measure.compProd_apply (MeasurableEquiv.prodAssoc.measurable hs),\n Measure.lintegral_compProd]\n swap; \u00b7 exact measurable_kernel_prod_mk_left (MeasurableEquiv.prodAssoc.measurable hs)\n congr with a\n rw [compProd_apply _ _ _ (measurable_prod_mk_left hs)]\n congr\n\nlemma Measure.compProd_compProd' (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n (\u03be : kernel T S) [IsMarkovKernel \u03be] (\u03ba : kernel (T \u00d7 S) U) [IsMarkovKernel \u03ba] :\n \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba) = (\u03bc \u2297\u2098 \u03be \u2297\u2098 \u03ba).comap\n (MeasurableEquiv.prodAssoc.symm : T \u00d7 S \u00d7 U \u2243\u1d50 (T \u00d7 S) \u00d7 U) := by\n rw [MeasurableEquiv.comap_symm, Measure.compProd_compProd]\n\nlemma Measure.compProd_compProd'' (\u03bc : Measure T) [IsProbabilityMeasure \u03bc]\n (\u03be : kernel T S) [IsMarkovKernel \u03be] (\u03ba : kernel (T \u00d7 S) U) [IsMarkovKernel \u03ba] :\n \u03bc \u2297\u2098 \u03be \u2297\u2098 \u03ba = Measure.comap MeasurableEquiv.prodAssoc (\u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)) := by\n rw [Measure.compProd_compProd, \u2190 MeasurableEquiv.map_symm, Measure.map_map]\n \u00b7 simp\n \u00b7 exact MeasurableEquiv.prodAssoc.symm.measurable\n \u00b7 exact MeasurableEquiv.prodAssoc.measurable\n\n-- from kernel (T \u00d7 S \u00d7 U) V ; Measure (T \u00d7 S \u00d7 U)\n-- to kernel (T \u00d7 S) V ; Measure (T \u00d7 S)\nlemma entropy_submodular_compProd {\u03be : kernel T S} [IsMarkovKernel \u03be]\n {\u03ba : kernel (T \u00d7 S) U} [IsMarkovKernel \u03ba] {\u03b7 : kernel (T \u00d7 S \u00d7 U) V} [IsMarkovKernel \u03b7]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba (\u03bc \u2297\u2098 \u03be))\n (h\u03b7 : AEFiniteKernelSupport \u03b7 (\u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba))) (h\u03be : AEFiniteKernelSupport \u03be \u03bc) :\n Hk[\u03b7, \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)]\n \u2264 Hk[snd (\u03ba \u2297\u2096 (comap \u03b7 MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable)), \u03bc \u2297\u2098 \u03be] := by\n have h_meas := (MeasurableEquiv.prodAssoc.symm : T \u00d7 S \u00d7 U \u2243\u1d50 (T \u00d7 S) \u00d7 U).symm.measurable\n have : FiniteSupport (\u03bc \u2297\u2098 \u03be) := finiteSupport_of_compProd h\u03be\n have : FiniteSupport (\u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)) := finiteSupport_of_compProd (h\u03be.compProd h\u03ba)\n have h := entropy_condKernel_le_entropy_snd\n (\u03ba := \u03ba \u2297\u2096 (comap \u03b7 MeasurableEquiv.prodAssoc h_meas)) (\u03bc := \u03bc \u2297\u2098 \u03be) ?_\n simp only [fst_compProd] at h\n have : condKernel (\u03ba \u2297\u2096 comap \u03b7 \u2191MeasurableEquiv.prodAssoc h_meas)\n =\u1d50[\u03bc \u2297\u2098 \u03be \u2297\u2098 \u03ba] comap \u03b7 \u2191MeasurableEquiv.prodAssoc h_meas := by\n exact condKernel_compProd_ae_eq \u03ba (comap \u03b7 \u2191MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable)\n (\u03bc \u2297\u2098 \u03be)\n rw [entropy_congr this, Measure.compProd_compProd'', entropy_comap_equiv] at h\n . exact h\n \u00b7 refine (h\u03ba.compProd ?_)\n convert h\u03b7.comap_equiv MeasurableEquiv.prodAssoc\n exact Measure.compProd_compProd'' _ _ _\n\nlemma entropy_condKernel_compProd_triple (\u03be : kernel T S) [IsMarkovKernel \u03be]\n (\u03ba : kernel (T \u00d7 S) U) [IsMarkovKernel \u03ba] (\u03b7 : kernel (T \u00d7 S \u00d7 U) V) [IsMarkovKernel \u03b7]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] :\n Hk[condKernel (\u03be \u2297\u2096 \u03ba \u2297\u2096 \u03b7) , \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)] = Hk[\u03b7, \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)] :=\n entropy_congr (condKernel_compProd_ae_eq (\u03be \u2297\u2096 \u03ba) \u03b7 \u03bc)\n\n/- $$ H[X,Y,Z] + H[X] \\leq H[Z,X] + H[Y,X].$$ -/\nlemma entropy_compProd_triple_add_entropy_le {\u03be : kernel T S} [IsMarkovKernel \u03be]\n {\u03ba : kernel (T \u00d7 S) U} [IsMarkovKernel \u03ba] {\u03b7 : kernel (T \u00d7 S \u00d7 U) V} [IsMarkovKernel \u03b7]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba (\u03bc \u2297\u2098 \u03be))\n (h\u03b7 : AEFiniteKernelSupport \u03b7 (\u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba))) (h\u03be : AEFiniteKernelSupport \u03be \u03bc) :\n Hk[(\u03be \u2297\u2096 \u03ba) \u2297\u2096 \u03b7, \u03bc] + Hk[\u03be, \u03bc]\n \u2264 Hk[\u03be \u2297\u2096 snd (\u03ba \u2297\u2096 comap \u03b7 MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable), \u03bc]\n + Hk[\u03be \u2297\u2096 \u03ba, \u03bc] := by\n rw [chain_rule,\n chain_rule (\u03ba := \u03be \u2297\u2096 snd (\u03ba \u2297\u2096 comap \u03b7 \u2191MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable))]\n simp only [fst_compProd, entropy_condKernel_compProd_triple, fst_deleteMiddle]\n . calc Hk[\u03be \u2297\u2096 \u03ba , \u03bc] + Hk[\u03b7 , \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)] + Hk[\u03be , \u03bc]\n = Hk[\u03be , \u03bc] + Hk[\u03be \u2297\u2096 \u03ba , \u03bc] + Hk[\u03b7 , \u03bc \u2297\u2098 (\u03be \u2297\u2096 \u03ba)] := by abel\n _ \u2264 Hk[\u03be , \u03bc] + Hk[\u03be \u2297\u2096 \u03ba , \u03bc]\n + Hk[condKernel (\u03be \u2297\u2096 snd (\u03ba \u2297\u2096 comap \u03b7 MeasurableEquiv.prodAssoc _)) , \u03bc \u2297\u2098 \u03be] := by\n refine add_le_add le_rfl ?_\n refine (entropy_submodular_compProd h\u03ba h\u03b7 h\u03be).trans_eq ?_\n refine entropy_congr ?_\n exact (condKernel_compProd_ae_eq _ _ _).symm\n _ = Hk[\u03be , \u03bc] + Hk[condKernel (\u03be \u2297\u2096 snd (\u03ba \u2297\u2096 comap \u03b7 MeasurableEquiv.prodAssoc _)) , \u03bc \u2297\u2098 \u03be] + Hk[\u03be \u2297\u2096 \u03ba , \u03bc] := by abel\n . refine h\u03be.compProd ?_\n refine AEFiniteKernelSupport.snd ?_\n refine h\u03ba.compProd ?_\n convert h\u03b7.comap_equiv MeasurableEquiv.prodAssoc\n exact Measure.compProd_compProd'' _ _ _\n \u00b7 exact (h\u03be.compProd h\u03ba).compProd h\u03b7\n\n/-- The submodularity inequality:\n$$ H[X,Y,Z] + H[X] \\leq H[X,Z] + H[X,Y].$$ -/\nlemma entropy_triple_add_entropy_le' {\u03ba : kernel T (S \u00d7 U \u00d7 V)} [IsMarkovKernel \u03ba]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[\u03ba, \u03bc] + Hk[fst \u03ba, \u03bc] \u2264 Hk[deleteMiddle \u03ba, \u03bc] + Hk[deleteRight \u03ba, \u03bc] := by\n set \u03ba' := map \u03ba MeasurableEquiv.prodAssoc.symm MeasurableEquiv.prodAssoc.symm.measurable\n with h\u03ba'_def\n let \u03be := fst (fst \u03ba')\n let \u03ba'' := condKernel (fst \u03ba')\n let \u03b7 := condKernel \u03ba'\n have h\u03be_eq : \u03be = fst \u03ba := by\n simp only [\u03be, \u03ba', fst, MeasurableEquiv.prodAssoc, MeasurableEquiv.coe_mk, Equiv.coe_fn_mk,\n map_map]\n congr\n have h_compProd_eq : \u03be \u2297\u2096 \u03ba'' = fst \u03ba' := (disintegration (fst \u03ba')).symm\n have h_compProd_triple_eq : (\u03be \u2297\u2096 \u03ba'') \u2297\u2096 \u03b7 = \u03ba' := by\n rw [h_compProd_eq]\n exact (disintegration \u03ba').symm\n have h_compProd_triple_eq' : \u03be \u2297\u2096 (\u03ba'' \u2297\u2096 comap \u03b7 MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable)\n = \u03ba := by\n rw [\u2190 compProd_assoc, h_compProd_triple_eq,h\u03ba'_def, map_map]\n simp\n have h := entropy_compProd_triple_add_entropy_le (\u03be := \u03be) (\u03ba := \u03ba'') (\u03b7 := \u03b7) (\u03bc := \u03bc) ?_ ?_ ?_\n rotate_left\n . exact aefiniteKernelSupport_of_cond _ (h\u03ba.map _).fst\n . rw [h_compProd_eq]\n apply aefiniteKernelSupport_of_cond\n exact h\u03ba.map _\n \u00b7 exact (h\u03ba.map _).fst.fst\n rw [\u2190 h\u03be_eq]\n have h_right : deleteRight \u03ba = fst \u03ba' := by\n simp only [\u03ba', deleteRight, fst, map_map]\n congr\n have h_middle : deleteMiddle \u03ba\n = \u03be \u2297\u2096 snd (\u03ba'' \u2297\u2096 comap \u03b7 MeasurableEquiv.prodAssoc MeasurableEquiv.prodAssoc.measurable) := by\n rw [\u2190 deleteMiddle_compProd, h_compProd_triple_eq']\n have h\u03ba : Hk[\u03ba, \u03bc] = Hk[\u03ba', \u03bc] := by\n rw [h\u03ba'_def, entropy_map_of_injective]\n exact MeasurableEquiv.prodAssoc.symm.injective\n rw [h_right, h_middle, h\u03ba, \u2190 h_compProd_triple_eq, fst_compProd]\n . exact h\n\nlemma entropy_reverse {\u03ba : kernel T (S \u00d7 U \u00d7 V)} [IsMarkovKernel \u03ba]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[reverse \u03ba, \u03bc] = Hk[\u03ba, \u03bc] := by\n refine le_antisymm ?_ ?_\n \u00b7 convert entropy_map_le (fun p \u21a6 (p.2.2, p.2.1, p.1)) h\u03ba\n \u00b7 conv_lhs => rw [\u2190 reverse_reverse \u03ba]\n convert entropy_map_le (\u03ba := reverse \u03ba) (fun p \u21a6 (p.2.2, p.2.1, p.1)) h\u03ba.reverse\n\n", "theoremStatement": "/-- The submodularity inequality:\n$$ H[X,Y,Z] + H[Z] \\leq H[X,Z] + H[Y,Z].$$ -/\nlemma entropy_triple_add_entropy_le (\u03ba : kernel T (S \u00d7 U \u00d7 V)) [IsMarkovKernel \u03ba]\n (\u03bc : Measure T) [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[\u03ba, \u03bc] + Hk[snd (snd \u03ba), \u03bc] \u2264 Hk[deleteMiddle \u03ba, \u03bc] + Hk[snd \u03ba, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_triple_add_entropy_le", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "0a4b141ca334c16e17a6ca547295886415d54340", "date": "2023-11-15"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/MutualInfo.lean", "module": "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.MutualInfo.jsonl", "positionMetadata": {"lineInFile": 311, 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"Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.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.Lemma", "Mathlib.Tactic.TypeStar", "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.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.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.Init.Data.Int.Basic", "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.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "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.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.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.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "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.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.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h2 : fst (reverse \u03ba) = snd (snd \u03ba) := by\n simp only [fst, reverse, snd, map_map]\n congr\n rw [\u2190 entropy_reverse h\u03ba, \u2190 h2]\n refine (entropy_triple_add_entropy_le' (\u03ba := reverse \u03ba) (\u03bc:= \u03bc) h\u03ba.reverse).trans ?_\n refine add_le_add ?_ ?_\n \u00b7 rw [\u2190 entropy_swapRight]\n simp\n \u00b7 rw [\u2190 entropy_swapRight]\n simp", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 329}} +{"srcContext": "import Mathlib.Combinatorics.Additive.Energy\nimport Mathlib.Analysis.NormedSpace.PiLp\nimport Mathlib.Analysis.InnerProductSpace.PiL2\nimport LeanAPAP.Extras.BSG\nimport PFR.HomPFR\n\n/-!\n# The approximate homomorphism form of PFR\n\nHere we apply PFR to show that almost homomorphisms f from a 2-group to a 2-group often coincide with a (shifted) actual homomorphisms. Here, approximate is in the sense that f(x+y)=f(x)+f(y) is true for a positive proportion of x,y.\n\n## Main result\n\n* `approx_hom_pfr` : If $f: G \u2192 G'$ is a map between finite abelian elementary 2-groups such that\n $f(x+y)=f(x)+f(y)$ for at least $|G|/K$ values, then then there is a homomorphism $\\phi: G \\to G'$\n and a constant $c$ such that $f(x)=\\phi(x)+c$ for a substantial set of values.\n-/\n\nopen Finset\nopen scoped BigOperators Classical Pointwise\n\nvariable {G G' : Type*} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G']\n [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (A : Finset G)\n\nset_option maxHeartbeats 400000 in\n", "theoremStatement": "/-- Let $G, G'$ be finite abelian $2$-groups.\nLet $f : G \\to G'$ be a function, and suppose that there are at least\n$|G|^2 / K$ pairs $(x,y) \\in G^2$ such that $$ f(x+y) = f(x) + f(y).$$\nThen there exists a homomorphism $\\phi : G \\to G'$ and a constant $c \\in G'$ such that\n$f(x) = \\phi(x)+c$ for at least $|G| / (2 ^ {172} * K ^ {146})$ values of $x \\in G$. -/\ntheorem approx_hom_pfr (f : G \u2192 G') (K : \u211d) (hK : K > 0)\n (hf : Nat.card {x : G \u00d7 G | f (x.1 + x.2) = f x.1 + f x.2} \u2265 Nat.card G ^ 2 / K) :\n \u2203 (\u03c6 : G \u2192+ G') (c : G'), Nat.card {x | f x = \u03c6 x + c} \u2265 Nat.card G / (2 ^ 172 * K ^ 146) ", "theoremName": "approx_hom_pfr", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "bdec9e981658631f456b4d9a321a157b61970374", "date": "2023-12-09"}, "file": "PFR/PFR/ApproxHomPFR.lean", "module": "PFR.ApproxHomPFR", "jsonFile": "PFR.ApproxHomPFR.jsonl", "positionMetadata": {"lineInFile": 26, "tokenPositionInFile": 1017, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 535, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", 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"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.Paths", "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", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "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", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "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", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "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", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Fintype.Prod", "Mathlib.Combinatorics.Additive.Energy", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.Algebra.Module.BigOperators", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Extr", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "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.Reindex", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "LeanAPAP.Mathlib.Combinatorics.Additive.Energy", "LeanAPAP.Mathlib.Tactic.Positivity", "LeanAPAP.Mathlib.Data.Fintype.Pi", "Mathlib.Tactic.Positivity.Finset", "LeanAPAP.Mathlib.Data.NNRat.Defs", "LeanAPAP.Prereqs.NNRat.Defs", "Mathlib.Data.NNRat.Lemmas", "LeanAPAP.Prereqs.NNRat.Cast.Defs", "LeanAPAP.Prereqs.NNRat.Cast.CharZero", "LeanAPAP.Prereqs.NNRat.Algebra", "LeanAPAP.Prereqs.NNRat.GroupPower.Lemmas", "LeanAPAP.Prereqs.Expect.Basic", "LeanAPAP.Prereqs.Translate", "LeanAPAP.Prereqs.Indicator", "LeanAPAP.Prereqs.Discrete.Convolution.Basic", "LeanAPAP.Prereqs.Discrete.Convolution.Order", "LeanAPAP.Extras.BSG", "Mathlib.Data.Set.Card", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main", "PFR.ImprovedPFR", "PFR.ForMathlib.Graph", "PFR.Mathlib.LinearAlgebra.Basis.VectorSpace", "PFR.Mathlib.SetTheory.Cardinal.Finite", "PFR.HomPFR"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let A := (Set.graph f).toFinite.toFinset\n\n have h_cs : ((A \u00d7\u02e2 A).filter (fun (a, a') \u21a6 a + a' \u2208 A) |>.card : \u211d) ^ 2 \u2264\n Finset.card A * E[A] := by\n norm_cast\n convert card_sq_le_card_mul_additiveEnergy A A A\n rewrite [\u2190 Nat.card_eq_finsetCard, \u2190 Nat.card_eq_finsetCard,\n Nat.card_congr (Set.equiv_filter_graph f)] at h_cs\n\n have hA : Nat.card A = Nat.card G := by\n rewrite [\u2190 Set.card_graph f, Nat.card_eq_finsetCard, Set.Finite.card_toFinset]; simp\n have hA_pos : 0 < (Nat.card A : \u211d) := Nat.cast_pos.mpr <| Nat.card_pos.trans_eq hA.symm\n have : ((Nat.card G)^2 / K)^2 \u2264 Nat.card A * E[A] := LE.le.trans (by gcongr) h_cs\n rewrite [\u2190 hA] at this\n replace : E[A] \u2265 (Finset.card A)^3 / K^2 := calc\n _ \u2265 ((Nat.card A)^2 / K)^2 / Nat.card A := (div_le_iff' <| hA_pos).mpr this\n _ = ((Nat.card A)^4 / (Nat.card A)) / K^2 := by ring\n _ = (Finset.card A)^3 / K^2 := by\n rw [pow_succ, mul_div_assoc, div_self (ne_of_gt hA_pos), mul_one,\n Nat.card_eq_finsetCard]\n have hA_nonempty : A.Nonempty := (Set.Finite.toFinset_nonempty _).2 $ Set.graph_nonempty _\n obtain \u27e8A', hA', hA'1, hA'2\u27e9 :=\n BSG_self' (sq_nonneg K) hA_nonempty (by simpa only [inv_mul_eq_div] using this)\n clear h_cs hf this\n have hA'\u2080 : A'.Nonempty := Finset.card_pos.1 $ Nat.cast_pos.1 $ hA'1.trans_lt' $ by\n have := hA_nonempty.card_pos; positivity\n\n let A'' := A'.toSet\n have hA''_coe : Nat.card A'' = Finset.card A' := Nat.card_eq_finsetCard A'\n have hA''_pos : 0 < Nat.card A'' := by rw [hA''_coe]; exact hA'\u2080.card_pos\n have hA''_nonempty : Set.Nonempty A'' := nonempty_subtype.mp (Finite.card_pos_iff.mp hA''_pos)\n have : Finset.card (A' - A') = Nat.card (A'' + A'') := calc\n _ = Nat.card (A' - A').toSet := (Nat.card_eq_finsetCard _).symm\n _ = Nat.card (A'' + A'') := by rw [Finset.coe_sub, sumset_eq_sub]\n replace : Nat.card (A'' + A'') \u2264 2 ^ 14 * K ^ 12 * Nat.card A'' := by\n rewrite [\u2190 this, hA''_coe]\n simpa [\u2190 pow_mul] using hA'2\n obtain \u27e8H, c, hc_card, hH_le, hH_ge, hH_cover\u27e9 := PFR_conjecture_improv_aux hA''_nonempty this\n clear hA'2 hA''_coe hH_le hH_ge hA_pos\n obtain \u27e8H\u2080, H\u2081, \u03c6, hH\u2080H\u2081, hH\u2080H\u2081_card\u27e9 := goursat H\n\n have h_le_H\u2080 : Nat.card A'' \u2264 Nat.card c * Nat.card H\u2080 := by\n have h_le := Nat.card_mono (Set.toFinite _) (Set.image_subset Prod.fst hH_cover)\n have h_proj_A'' : Nat.card A'' = Nat.card (Prod.fst '' A'') := Nat.card_congr\n (Equiv.Set.imageOfInjOn Prod.fst A'' <|\n Set.InjOn.mono (Set.Finite.subset_toFinset.mp hA') (Set.fst_injOn_graph f))\n have h_proj_c : Prod.fst '' (c + H : Set (G \u00d7 G')) = (Prod.fst '' c) + H\u2080 := by\n ext x ; constructor <;> intro hx\n \u00b7 obtain \u27e8x, \u27e8\u27e8c, hc, h, hh, hch\u27e9, hx\u27e9\u27e9 := hx\n rewrite [\u2190 hx]\n exact \u27e8c.1, Set.mem_image_of_mem Prod.fst hc, h.1, ((hH\u2080H\u2081 h).mp hh).1, (Prod.ext_iff.mp hch).1\u27e9\n \u00b7 obtain \u27e8_, \u27e8c, hc\u27e9, h, hh, hch\u27e9 := hx\n refine \u27e8c + (h, \u03c6 h), \u27e8\u27e8c, hc.1, (h, \u03c6 h), ?_\u27e9, by rwa [\u2190 hc.2] at hch\u27e9\u27e9\n exact \u27e8(hH\u2080H\u2081 \u27e8h, \u03c6 h\u27e9).mpr \u27e8hh, by rw [sub_self]; apply zero_mem\u27e9, rfl\u27e9\n rewrite [\u2190 h_proj_A'', h_proj_c] at h_le\n apply (h_le.trans Set.card_add_le).trans\n gcongr\n exact Nat.card_image_le c.toFinite\n\n have hH\u2080_pos : (0 : \u211d) < Nat.card H\u2080 := Nat.cast_pos.mpr Nat.card_pos\n have h_le_H\u2081 : (Nat.card H\u2081 : \u211d) \u2264 (Nat.card c) * (Nat.card H) / Nat.card A'' := calc\n _ = (Nat.card H : \u211d) / (Nat.card H\u2080) :=\n (eq_div_iff <| ne_of_gt <| hH\u2080_pos).mpr <| by rw [mul_comm, \u2190 Nat.cast_mul, hH\u2080H\u2081_card]\n _ \u2264 (Nat.card c : \u211d) * (Nat.card H) / Nat.card A'' := by\n nth_rewrite 1 [\u2190 mul_one (Nat.card H : \u211d), mul_comm (Nat.card c : \u211d)]\n repeat rewrite [mul_div_assoc]\n refine mul_le_mul_of_nonneg_left ?_ (Nat.cast_nonneg _)\n refine le_of_mul_le_mul_right ?_ hH\u2080_pos\n refine le_of_mul_le_mul_right ?_ (Nat.cast_pos.mpr hA''_pos)\n rewrite [div_mul_cancel\u2080 1, mul_right_comm, one_mul, div_mul_cancel\u2080, \u2190 Nat.cast_mul]\n \u00b7 exact Nat.cast_le.mpr h_le_H\u2080\n \u00b7 exact ne_of_gt (Nat.cast_pos.mpr hA''_pos)\n \u00b7 exact ne_of_gt hH\u2080_pos\n clear h_le_H\u2080 hA''_pos hH\u2080_pos hH\u2080H\u2081_card\n\n let translate (c : G \u00d7 G') (h : G') := A'' \u2229 ({c} + {(0, h)} + Set.graph \u03c6.toFun)\n have h_translate (c : G \u00d7 G') (h : G') :\n Prod.fst '' translate c h \u2286 { x : G | f x = \u03c6 x + (-\u03c6 c.1 + c.2 + h) } := by\n intro x hx\n obtain \u27e8x, \u27e8\u27e8hxA'', \u27e8_, \u27e8c', hc, h', hh, hch\u27e9, x', hx, hchx\u27e9\u27e9, hxx\u27e9\u27e9 := hx\n show f _ = \u03c6 _ + (-\u03c6 c.1 + c.2 + h)\n replace := (Set.mem_graph x).mp <| (Set.graph f).toFinite.mem_toFinset.mp (hA' hxA'')\n rewrite [\u2190 hxx, this, \u2190 hchx, \u2190 hch, hc, hh]\n show c.2 + h + x'.2 = \u03c6 (c.1 + 0 + x'.1) + (-\u03c6 c.1 + c.2 + h)\n replace : \u03c6 x'.1 = x'.2 := (Set.mem_graph x').mp hx\n rw [map_add, map_add, map_zero, add_zero, this, add_comm (\u03c6 c.1), add_assoc x'.2,\n \u2190 add_assoc (\u03c6 c.1), \u2190 add_assoc (\u03c6 c.1), add_neg_self, zero_add, add_comm]\n have h_translate_card c h : Nat.card (translate c h) = Nat.card (Prod.fst '' translate c h) :=\n Nat.card_congr (Equiv.Set.imageOfInjOn Prod.fst (translate c h) <|\n Set.InjOn.mono (fun _ hx \u21a6 Set.Finite.subset_toFinset.mp hA' hx.1) (Set.fst_injOn_graph f))\n\n let cH\u2081 := (c \u00d7\u02e2 H\u2081).toFinite.toFinset\n haveI A_nonempty : Nonempty A'' := Set.nonempty_coe_sort.mpr hA''_nonempty\n replace : Nonempty c := by\n obtain \u27e8c, hc, _, _, _\u27e9 := hH_cover (Classical.choice A_nonempty).property\n exact \u27e8c, hc\u27e9\n replace h_nonempty : Set.Nonempty (c \u00d7\u02e2 H\u2081) :=\n Set.Nonempty.prod (Set.nonempty_coe_sort.mp this) (Set.nonempty_coe_sort.mp inferInstance)\n replace : A' = Finset.biUnion cH\u2081 fun ch \u21a6 (translate ch.1 ch.2).toFinite.toFinset := by\n ext x ; constructor <;> intro hx\n \u00b7 obtain \u27e8c', hc, h, hh, hch\u27e9 := hH_cover hx\n refine Finset.mem_biUnion.mpr \u27e8(c', h.2 - \u03c6 h.1), ?_\u27e9\n refine \u27e8(Set.Finite.mem_toFinset _).mpr \u27e8hc, ((hH\u2080H\u2081 h).mp hh).2\u27e9, ?_\u27e9\n refine (Set.Finite.mem_toFinset _).mpr \u27e8hx, c' + (0, h.2 - \u03c6 h.1), ?_\u27e9\n refine \u27e8\u27e8c', rfl, (0, h.2 - \u03c6 h.1), rfl, rfl\u27e9, (h.1, \u03c6 h.1), \u27e8h.1, rfl\u27e9, ?_\u27e9\n beta_reduce\n rewrite [add_assoc]\n show c' + (0 + h.1, h.2 - \u03c6 h.1 + \u03c6 h.1) = x\n rewrite [zero_add, sub_add_cancel]\n exact hch\n \u00b7 obtain \u27e8ch, hch\u27e9 := Finset.mem_biUnion.mp hx\n exact ((Set.Finite.mem_toFinset _).mp hch.2).1\n\n replace : \u2211 __ in cH\u2081, ((2 ^ 4)\u207b\u00b9 * (K ^ 2)\u207b\u00b9 * A.card / cH\u2081.card : \u211d) \u2264\n \u2211 ch in cH\u2081, ((translate ch.1 ch.2).toFinite.toFinset.card : \u211d) := by\n rewrite [Finset.sum_const, nsmul_eq_mul, \u2190 mul_div_assoc, mul_div_right_comm, div_self, one_mul]\n \u00b7 apply hA'1.trans\n norm_cast\n exact (congrArg Finset.card this).trans_le Finset.card_biUnion_le\n \u00b7 symm\n refine ne_of_lt <| Nat.cast_zero.symm.trans_lt <| Nat.cast_lt.mpr <| Finset.card_pos.mpr ?_\n exact (Set.Finite.toFinset_nonempty _).mpr h_nonempty\n replace : \u2203 c' : G \u00d7 G', \u2203 h : G', (2 ^ 4 : \u211d)\u207b\u00b9 * (K ^ 2)\u207b\u00b9 * A.card / cH\u2081.card \u2264\n Nat.card { x : G | f x = \u03c6 x + (-\u03c6 c'.1 + c'.2 + h) } := by\n obtain \u27e8ch, hch\u27e9 :=\n Finset.exists_le_of_sum_le ((Set.Finite.toFinset_nonempty _).mpr h_nonempty) this\n refine \u27e8ch.1, ch.2, hch.2.trans ?_\u27e9\n rewrite [Set.Finite.card_toFinset, \u2190 Nat.card_eq_fintype_card, h_translate_card]\n exact Nat.cast_le.mpr <| Nat.card_mono (Set.toFinite _) (h_translate ch.1 ch.2)\n clear hA' hA'1 hH_cover hH\u2080H\u2081 translate h_translate h_translate_card\n\n obtain \u27e8c', h, hch\u27e9 := this\n use \u03c6, -\u03c6 c'.1 + c'.2 + h\n refine le_trans ?_ hch\n unfold_let cH\u2081\n rewrite [\u2190 Nat.card_eq_finsetCard, hA, \u2190 mul_inv, inv_mul_eq_div, div_div]\n apply div_le_div (Nat.cast_nonneg _) le_rfl\n \u00b7 apply mul_pos\n \u00b7 apply mul_pos\n \u00b7 norm_num\n \u00b7 exact sq_pos_of_pos hK\n \u00b7 rewrite [Nat.cast_pos, Finset.card_pos, Set.Finite.toFinset_nonempty _]\n exact h_nonempty\n rw [show 146 = 2 + 144 by norm_num, show 172 = 4 + 168 by norm_num, pow_add, pow_add,\n mul_mul_mul_comm]\n gcongr\n rewrite [\u2190 c.toFinite.toFinset_prod (H\u2081 : Set G').toFinite, Finset.card_product]\n repeat rewrite [Set.Finite.card_toFinset, \u2190 Nat.card_eq_fintype_card]\n push_cast\n refine (mul_le_mul_of_nonneg_left h_le_H\u2081 (Nat.cast_nonneg _)).trans ?_\n refine (mul_le_mul_of_nonneg_right hc_card (by positivity)).trans ?_\n rewrite [mul_div_left_comm, mul_assoc]\n refine (mul_le_mul_of_nonneg_right hc_card (by positivity)).trans_eq ?_\n rw [mul_assoc ((_ * _)^6), mul_mul_mul_comm, mul_comm (_ ^ (1/2) * _), mul_comm_div,\n \u2190 mul_assoc _ (_^_) (_^_), mul_div_assoc, mul_mul_mul_comm _ (_^_) (_^_),\n \u2190 mul_div_assoc, mul_assoc _ (_^(1/2)) (_^(1/2)),\n \u2190 Real.rpow_add (Nat.cast_pos.mpr Nat.card_pos), add_halves, Real.rpow_one,\n \u2190 Real.rpow_add (Nat.cast_pos.mpr Nat.card_pos), add_halves, Real.rpow_neg_one,\n mul_comm _ (_ / _), mul_assoc (_^6)]\n conv => { lhs; rhs; rw [\u2190 mul_assoc, \u2190 mul_div_assoc, mul_comm_div, mul_div_assoc] }\n rw [div_self <| Nat.cast_ne_zero.mpr (Nat.ne_of_lt Nat.card_pos).symm, mul_one]\n push_cast\n rw [mul_inv_cancel <| Nat.cast_ne_zero.mpr (Nat.ne_of_lt Nat.card_pos).symm, one_mul, \u2190 sq,\n \u2190 Real.rpow_two, \u2190 Real.rpow_mul (by positivity), Real.mul_rpow (by positivity) (by positivity)]\n have : K ^ (12 : \u2115) = K ^ (12 : \u211d) := (Real.rpow_nat_cast K 12).symm\n rw [this, \u2190 Real.rpow_mul (by positivity)]\n norm_num\n exact Real.rpow_nat_cast K 144", "proofType": "tactic", "proofLengthLines": 168, "proofLengthTokens": 9237}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] := by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]\n\n/-- Adding a constant to a random variable does not change the Rusza distance. -/\nlemma rdist_add_const [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y + fun _ \u21a6 c; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n have A : IdentDistrib (Y' + fun _ \u21a6 c) (Y + fun _ \u21a6 c) \u03bd \u03bc' := by\n change IdentDistrib (fun \u03c9 \u21a6 Y' \u03c9 + c) (fun \u03c9 \u21a6 Y \u03c9 + c) \u03bd \u03bc'\n apply hIdY.comp (measurable_add_const c)\n have B : IndepFun X' (Y' + fun _ \u21a6 c) \u03bd := by\n change IndepFun X' (fun \u03c9 \u21a6 Y' \u03c9 + c) \u03bd\n apply hind.comp measurable_id (measurable_add_const c)\n have C : X' - (Y' + fun _ \u21a6 c) = (X' - Y') + (fun _ \u21a6 -c) := by\n ext \u03c9; simp; abel\n rw [\u2190 hIdX.rdist_eq hIdY, \u2190 hIdX.rdist_eq A, hind.rdist_eq hX' hY',\n B.rdist_eq hX' (hY'.add_const _), entropy_add_const hY' c, C, entropy_add_const]\n exact hX'.sub hY'\n\n/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\nlemma rdist_add_const' [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] (c: G) (c': G)\n (hX : Measurable X) (hY : Measurable Y) :\n d[X + fun _ \u21a6 c; \u03bc # Y + fun _ \u21a6 c'; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_add_const _ hY, rdist_symm, rdist_add_const hY hX, rdist_symm]\n measurability\n\n", "theoremStatement": "/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (X : \u03a9 \u2192 G) (Y : \u03a9 \u2192 G) (Z : \u03a9 \u2192 G)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] ", "theoremName": "ent_of_diff_le", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "61fbca1a5afcf0aede80c8e0f215fe0488b6c455", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 355, "tokenPositionInFile": 18615, "theoremPositionInFile": 23}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 189, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", 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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", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have h1 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] + H[X - Y; \u03bc] \u2264 H[\u27e8X - Z, X - Y\u27e9; \u03bc] + H[\u27e8Y, X - Y\u27e9; \u03bc] :=\n entropy_triple_add_entropy_le \u03bc (hX.sub hZ) hY (hX.sub hY)\n have h2 : H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n calc H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n have : \u27e8X - Z, X - Y\u27e9 = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 \u27e8X - Z, Y - Z\u27e9 := by ext1; simp\n rw [this]\n apply entropy_comp_le \u03bc (by measurability)\n _ \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n have h : 0 \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] - H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n apply mutualInfo_nonneg (by measurability) (by measurability) \u03bc\n linarith\n have h3 : H[\u27e8Y, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X, Y\u27e9 ; \u03bc] := by\n have : \u27e8Y, X - Y\u27e9 = (fun p \u21a6 (p.2, p.1 - p.2)) \u2218 \u27e8X, Y\u27e9 := by ext1; simp\n rw [this]\n exact entropy_comp_le \u03bc (hX.prod_mk hY) _\n have h4 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] = H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] := by\n refine entropy_of_comp_eq_of_comp \u03bc ((hX.sub hZ).prod_mk (hY.prod_mk (hX.sub hY)))\n (hX.prod_mk (hY.prod_mk hZ))\n (fun p : G \u00d7 (G \u00d7 G) \u21a6 (p.2.2 + p.2.1, p.2.1, -p.1 + p.2.2 + p.2.1))\n (fun p : G \u00d7 G \u00d7 G \u21a6 (p.1 - p.2.2, p.2.1, p.1 - p.2.1)) ?_ ?_\n \u00b7 ext1; simp\n \u00b7 ext1; simp\n have h5 : H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [entropy_assoc hX hY hZ, entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h\n rw [h4, h5] at h1\n calc H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Y - Z; \u03bc] - H[Z; \u03bc] := by linarith\n _ = H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n congr 2\n rw [entropy_sub_comm hY hZ]", "proofType": "tactic", "proofLengthLines": 30, "proofLengthTokens": 1567}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] := by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]\n\n/-- Adding a constant to a random variable does not change the Rusza distance. -/\nlemma rdist_add_const [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y + fun _ \u21a6 c; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n have A : IdentDistrib (Y' + fun _ \u21a6 c) (Y + fun _ \u21a6 c) \u03bd \u03bc' := by\n change IdentDistrib (fun \u03c9 \u21a6 Y' \u03c9 + c) (fun \u03c9 \u21a6 Y \u03c9 + c) \u03bd \u03bc'\n apply hIdY.comp (measurable_add_const c)\n have B : IndepFun X' (Y' + fun _ \u21a6 c) \u03bd := by\n change IndepFun X' (fun \u03c9 \u21a6 Y' \u03c9 + c) \u03bd\n apply hind.comp measurable_id (measurable_add_const c)\n have C : X' - (Y' + fun _ \u21a6 c) = (X' - Y') + (fun _ \u21a6 -c) := by\n ext \u03c9; simp; abel\n rw [\u2190 hIdX.rdist_eq hIdY, \u2190 hIdX.rdist_eq A, hind.rdist_eq hX' hY',\n B.rdist_eq hX' (hY'.add_const _), entropy_add_const hY' c, C, entropy_add_const]\n exact hX'.sub hY'\n\n/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\nlemma rdist_add_const' [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] (c: G) (c': G)\n (hX : Measurable X) (hY : Measurable Y) :\n d[X + fun _ \u21a6 c; \u03bc # Y + fun _ \u21a6 c'; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_add_const _ hY, rdist_symm, rdist_add_const hY hX, rdist_symm]\n measurability\n\n/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (X : \u03a9 \u2192 G) (Y : \u03a9 \u2192 G) (Z : \u03a9 \u2192 G)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n have h1 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] + H[X - Y; \u03bc] \u2264 H[\u27e8X - Z, X - Y\u27e9; \u03bc] + H[\u27e8Y, X - Y\u27e9; \u03bc] :=\n entropy_triple_add_entropy_le \u03bc (hX.sub hZ) hY (hX.sub hY)\n have h2 : H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n calc H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n have : \u27e8X - Z, X - Y\u27e9 = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 \u27e8X - Z, Y - Z\u27e9 := by ext1; simp\n rw [this]\n apply entropy_comp_le \u03bc (by measurability)\n _ \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n have h : 0 \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] - H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n apply mutualInfo_nonneg (by measurability) (by measurability) \u03bc\n linarith\n have h3 : H[\u27e8Y, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X, Y\u27e9 ; \u03bc] := by\n have : \u27e8Y, X - Y\u27e9 = (fun p \u21a6 (p.2, p.1 - p.2)) \u2218 \u27e8X, Y\u27e9 := by ext1; simp\n rw [this]\n exact entropy_comp_le \u03bc (hX.prod_mk hY) _\n have h4 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] = H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] := by\n refine entropy_of_comp_eq_of_comp \u03bc ((hX.sub hZ).prod_mk (hY.prod_mk (hX.sub hY)))\n (hX.prod_mk (hY.prod_mk hZ))\n (fun p : G \u00d7 (G \u00d7 G) \u21a6 (p.2.2 + p.2.1, p.2.1, -p.1 + p.2.2 + p.2.1))\n (fun p : G \u00d7 G \u00d7 G \u21a6 (p.1 - p.2.2, p.2.1, p.1 - p.2.1)) ?_ ?_\n \u00b7 ext1; simp\n \u00b7 ext1; simp\n have h5 : H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [entropy_assoc hX hY hZ, entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h\n rw [h4, h5] at h1\n calc H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Y - Z; \u03bc] - H[Z; \u03bc] := by linarith\n _ = H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n congr 2\n rw [entropy_sub_comm hY hZ]\n\n/-- The **entropic Ruzsa triangle inequality** -/\nlemma rdist_triangle {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [h\u03bc : IsProbabilityMeasure \u03bc] [h\u03bc' : IsProbabilityMeasure \u03bc'] [h\u03bc'' : IsProbabilityMeasure \u03bc''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n d[X ; \u03bc # Z ; \u03bc''] \u2264 d[X ; \u03bc # Y ; \u03bc'] + d[Y ; \u03bc' # Z ; \u03bc''] := by\n obtain \u27e8A, mA, \u03bcA, X', Y', Z', h\u03bcA, hInd, hX', hY', hZ', HX, HY, HZ, _, _, _\u27e9 :=\n independent_copies3_nondep_finiteRange hX hY hZ \u03bc \u03bc' \u03bc''\n suffices d[X' ; \u03bcA # Z' ; \u03bcA] \u2264 d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] by\n rwa [HX.rdist_eq HY, HY.rdist_eq HZ, HX.rdist_eq HZ] at this\n have IndepLem : IndepFun (\u27e8X', Z'\u27e9) Y' \u03bcA :=\n iIndepFun.indepFun_prod_mk hInd (fun i => by fin_cases i ; all_goals { simpa }) 0 2 1\n (by norm_cast) (by norm_cast)\n calc d[X' ; \u03bcA # Z' ; \u03bcA] = H[X' - Z'; \u03bcA] - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) := by\n rw [ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using hInd.indepFun (show 0 \u2260 2 by norm_cast)) hX' hZ'] ; ring\n _ \u2264 (H[X' - Y' ; \u03bcA] + H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA]) - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) :=\n sub_le_sub_right (ent_of_diff_le _ _ _ hX' hZ' hY' IndepLem) _\n _ = (H[X' - Y' ; \u03bcA] - H[X'; \u03bcA] / 2 - H[Y' ; \u03bcA] / 2) +\n (H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA] / 2 - H[Z'; \u03bcA] / 2) := by ring\n _ = d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] := by\n rw [ProbabilityTheory.IndepFun.rdist_eq (by simpa using (iIndepFun.indepFun hInd\n (show 0 \u2260 1 by norm_cast))) hX' hY', ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using (iIndepFun.indepFun hInd (show 1 \u2260 2 by norm_cast))) hY' hZ']\n\nvariable [MeasurableSingletonClass S] [MeasurableSingletonClass T]\n\n/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) [IsFiniteMeasure \u03bc]\n (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc'\"]\" => condRuzsaDist X Z Y W \u03bc \u03bc'\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" # \" Y \" | \" W \"]\" => condRuzsaDist X Z Y W volume volume\n\nlemma condRuzsaDist_def (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W] := rfl\n\n/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\nlemma condRuzsaDist_symm {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hZ : Measurable Z)\n (hW : Measurable W) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange Z]\n [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[Y | W ; \u03bc' # X | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist_symm]\n\n@[simp] lemma condRuszaDist_zero_right (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n d[X | Z ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero,\n kernel.rdist_zero_right]\n\n@[simp] lemma condRuszaDist_zero_left (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; 0 # Y | W ; \u03bc'] = 0 := by\n simp [condRuzsaDist]\n\nlemma condRuzsaDist_nonneg {X : \u03a9 \u2192 G} (hX : Measurable X) [FiniteRange X]\n {Z : \u03a9 \u2192 S} (hZ : Measurable Z) [FiniteRange Z]\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) [FiniteRange Y]\n {W : \u03a9' \u2192 T} (hW : Measurable W) [FiniteRange W]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n 0 \u2264 d[X | Z ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def]\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n refine kernel.rdist_nonneg ?_ ?_\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hX hZ\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hY hW\n\n/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\nlemma rdist_eq_rdistm : d[X ; \u03bc # Y ; \u03bc'] = kernel.rdistm (\u03bc.map X) (\u03bc'.map Y) := rfl\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\nlemma condRuzsaDist_eq_sum {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z in FiniteRange.toFinset Z, \u2211 w in FiniteRange.toFinset W,\n (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((((FiniteRange.toFinset Z)\n \u00d7\u02e2 (FiniteRange.toFinset W)) : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Finset.sum_product,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\nlemma condRuzsaDist_eq_sum' {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [Fintype S] [Fintype T] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z, \u2211 w, (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Fintype.sum_prod_type,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist' (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc' \"]\" => condRuzsaDist' X Y W \u03bc \u03bc'\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" # \" Y \" | \" W \"]\" => condRuzsaDist' X Y W volume volume\n\n/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\nlemma condRuzsaDist'_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y | W ; \u03bc'] =\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W] :=\n rfl\n\n@[simp] lemma condRuzsaDist'_zero_right (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) :\n d[X ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist'_def, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero, kernel.rdist_zero_right]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\nlemma condRuzsaDist'_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (dirac ()) (\u03bc'.map W) ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 FiniteRange.toFinset W : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, smul_eq_mul, Finset.sum_product]\n simp only [Finset.univ_unique, PUnit.default_eq_unit, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, Pi.one_apply, one_mul,\n Finset.sum_singleton]\n simp_rw [Measure.map_apply hW (measurableSet_singleton _)]\n congr with w\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff]\n refine Or.inr ?_\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm, condDistrib_apply hY hW _ _ hw]\n congr\n\n/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\nlemma condRuzsaDist'_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_eq_sum hY hW \u03bc \u03bc']\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\nopen scoped ENNReal\n\nlemma condRuzsaDist'_prod_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [FiniteRange W] [FiniteRange W']:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n have : d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc'] = \u2211 w in ((FiniteRange.toFinset W') \u00d7\u02e2 FiniteRange.toFinset W), (\u03bc' ((fun a => (W' a, W a)) \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; \u03bc'[|(fun a => (W' a, W a)) \u207b\u00b9' {w}]] := by\n rw [condRuzsaDist'_eq_sum hY (hW'.prod_mk hW)]\n apply Finset.sum_subset\n . intro (t, t')\n simp\n intro \u03c9 h1 h2\n exact \u27e8\u27e8\u03c9, h1\u27e9, \u27e8\u03c9, h2\u27e9\u27e9\n intro (t, t') _ ht\n simp at ht\n have : (fun a => (W' a, W a)) \u207b\u00b9' {(t, t')} = \u2205 := by\n ext \u03c9\n simp\n exact ht \u03c9\n simp [this]\n rw [this, Finset.sum_product_right]\n congr 1 with w\n rw [condRuzsaDist'_eq_sum hY hW', Finset.mul_sum]\n congr 1 with w'\n rw [\u2190 mul_assoc]\n have A : (fun a \u21a6 (W' a, W a)) \u207b\u00b9' {(w', w)} = W' \u207b\u00b9' {w'} \u2229 W\u207b\u00b9' {w} := by ext; simp\n congr 1\n \u00b7 simp only [ProbabilityTheory.cond, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n smul_eq_mul, ENNReal.toReal_mul, A, restrict_apply (hW' (MeasurableSet.singleton w'))]\n rcases eq_bot_or_bot_lt (\u03bc' (W \u207b\u00b9' {w})) with hw|hw\n \u00b7 have : \u03bc' (W' \u207b\u00b9' {w'} \u2229 W \u207b\u00b9' {w}) = 0 :=\n le_antisymm (le_trans (measure_mono (Set.inter_subset_right _ _)) hw.le) bot_le\n simp [hw, this]\n \u00b7 rw [\u2190 mul_assoc, \u2190 ENNReal.toReal_mul, ENNReal.mul_inv_cancel, ENNReal.one_toReal, one_mul]\n exacts [hw.ne', by finiteness]\n \u00b7 congr 1\n rw [A, cond_cond_eq_cond_inter' _ (hW (.singleton w)) (hW' (.singleton w')), Set.inter_comm]\n finiteness\n\n/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\nlemma condRuzsaDist'_prod_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [Fintype T]:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_prod_eq_sum \u03bc \u03bc' hY hW' hW]\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\nlemma condRuzsaDist'_eq_integral (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = (\u03bc'.map W)[fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])]] := by\n rw [condRuzsaDist'_eq_sum hY hW]\n simp_rw [\u2190 smul_eq_mul]\n have : (\u03bc'.map W) (FiniteRange.toFinset W : Set T)\u1d9c = 0 := by\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n convert symm $ integral_eq_sum' (\u03bc'.map W) this (fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])])\n rw [Measure.map_apply hW (MeasurableSet.singleton _)]\n\n/-- Conditioning by a constant does not affect Ruzsa distance. -/\nlemma condRuzsaDist_of_const {X : \u03a9 \u2192 G} (hX : Measurable X) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (c : S)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange W] :\n d[X|(fun _ \u21a6 c) ; \u03bc # Y | W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def, condRuzsaDist'_def, Measure.map_const,measure_univ,one_smul, kernel.rdist,\n kernel.rdist, integral_prod, integral_dirac, integral_prod,integral_dirac]\n dsimp; congr; ext x; congr\n rw [condDistrib_apply hX measurable_const]\n \u00b7 simp\n \u00b7 simp\n \u00b7 exact integrable_of_finiteSupport _\n \u00b7 exact integrable_of_finiteSupport _\n\n/-- If $(X,Z)$ and $(Y,W)$ are independent, then\n$$ d[X | Z ; Y | W] = H[X'- Y'|Z', W'] - H[X'|Z']/2 - H[Y'|W']/2$$\n-/\nlemma condRuzsaDist_of_indep\n {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun (\u27e8X, Z\u27e9) (\u27e8Y, W\u27e9) \u03bc) [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc] = H[X - Y | \u27e8Z, W\u27e9 ; \u03bc] - H[X | Z ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ (hZ.prod_mk hW),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hW]\n swap; \u00b7 exact hX.sub hY\n congr 2\n have hZW : IndepFun Z W \u03bc := by\n have h' := IndepFun.comp h measurable_snd measurable_snd\n exact h'\n have hZW_map : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map Z).prod (\u03bc.map W) :=\n (indepFun_iff_map_prod_eq_prod_map_map hZ.aemeasurable hW.aemeasurable).mp hZW\n rw [\u2190 hZW_map]\n refine kernel.entropy_congr ?_\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (hZ.prod_mk hW) _ _).symm\n refine (this.symm.trans ?_).symm\n suffices kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft S (condDistrib Y W \u03bc)\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc by\n filter_upwards [this] with x hx\n rw [kernel.map_apply, kernel.map_apply, hx]\n . exact (condDistrib_eq_prod_of_indepFun hX hZ hY hW \u03bc h).symm\n\n/-- Formula for conditional Ruzsa distance for independent sets of variables. -/\nlemma condRuzsaDist'_of_indep {X : \u03a9 \u2192 G} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun X (\u27e8Y, W\u27e9) \u03bc) [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc] = H[X - Y | W ; \u03bc] - H[X ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist'_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ hW,\n condEntropy_eq_kernel_entropy hY hW, entropy_eq_kernel_entropy]\n rotate_left\n \u00b7 exact hX.sub hY\n congr 2\n let Z : \u03a9 \u2192 Unit := fun _ \u21a6 ()\n rw [\u2190 condDistrib_unit_right hX \u03bc]\n have h' : IndepFun (\u27e8X,Z\u27e9) (\u27e8Y, W\u27e9) \u03bc := by\n rw [indepFun_iff_measure_inter_preimage_eq_mul]\n intro s t hs ht\n have : \u27e8X, Z\u27e9 \u207b\u00b9' s = X \u207b\u00b9' ((fun c \u21a6 (c, ())) \u207b\u00b9' s) := by ext1 y; simp\n rw [this]\n rw [indepFun_iff_measure_inter_preimage_eq_mul] at h\n exact h _ _ (measurable_prod_mk_right hs) ht\n have h_indep := condDistrib_eq_prod_of_indepFun hX measurable_const hY hW _ h'\n have h_meas_eq : \u03bc.map (\u27e8Z, W\u27e9) = (Measure.dirac ()).prod (\u03bc.map W) := by\n ext s hs\n rw [Measure.map_apply (measurable_const.prod_mk hW) hs, Measure.prod_apply hs, lintegral_dirac,\n Measure.map_apply hW (measurable_prod_mk_left hs)]\n congr\n rw [\u2190 h_meas_eq]\n have : kernel.map (kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft Unit (condDistrib Y W \u03bc)) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc)\n (fun x \u21a6 x.1 - x.2) measurable_sub := by\n filter_upwards [h_indep] with y hy\n conv_rhs => rw [kernel.map_apply, hy]\n rw [kernel.entropy_congr this]\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (measurable_const.prod_mk hW) _ _).symm\n rw [kernel.entropy_congr this]\n have h_meas : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map W).map (Prod.mk ()) := by\n ext s hs\n rw [Measure.map_apply measurable_prod_mk_left hs, h_meas_eq, Measure.prod_apply hs,\n lintegral_dirac]\n have h_ker : condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.prodMkLeft Unit (condDistrib (X - Y) W \u03bc) := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply (measurable_const.prod_mk hW) (measurableSet_singleton _)] at hx\n ext s hs\n have h_preimage_eq : (fun a \u21a6 (PUnit.unit, W a)) \u207b\u00b9' {x} = W \u207b\u00b9' {x.2} := by\n conv_lhs => rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod]\n ext1 y\n simp\n rw [kernel.prodMkLeft_apply, condDistrib_apply' _ (measurable_const.prod_mk hW) _ _ hx hs,\n condDistrib_apply' _ hW _ _ _ hs]\n rotate_left\n \u00b7 exact hX.sub hY\n \u00b7 convert hx\n exact h_preimage_eq.symm\n \u00b7 exact hX.sub hY\n congr\n rw [kernel.entropy_congr h_ker, h_meas, kernel.entropy_prodMkLeft_unit]\n\n/-- The conditional Ruzsa distance is unchanged if the sets of random variables are replaced with\ncopies. -/\nlemma condRuzsaDist_of_copy {X : \u03a9 \u2192 G} (hX : Measurable X) {Z : \u03a9 \u2192 S} (hZ : Measurable Z)\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) {W : \u03a9' \u2192 T} (hW : Measurable W)\n {X' : \u03a9'' \u2192 G} (hX' : Measurable X') {Z' : \u03a9'' \u2192 S} (hZ' : Measurable Z')\n {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib (\u27e8X, Z\u27e9) (\u27e8X', Z'\u27e9) \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange Z] [FiniteRange W] [FiniteRange Z'] [FiniteRange W'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[X' | Z' ; \u03bc'' # Y' | W' ; \u03bc'''] := by\n classical\n set A := (FiniteRange.toFinset Z) \u222a (FiniteRange.toFinset Z')\n set B := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n have hfull' : Measure.prod (\u03bc''.map Z') (\u03bc'''.map W') ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hZZ' : \u03bc.map Z = \u03bc''.map Z' := (h1.comp measurable_snd).map_eq\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hZZ', \u2190 hWW',\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hz : \u03bc (Z \u207b\u00b9' {x.1}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 have hZZ'x : \u03bc (Z \u207b\u00b9' {x.1}) = \u03bc'' (Z' \u207b\u00b9' {x.1}) := by\n have : \u03bc.map Z {x.1} = \u03bc''.map Z' {x.1} := by rw [hZZ']\n rwa [Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hZ' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hX hZ _ _ hz hs, condDistrib_apply' hX' hZ' _ _ _ hs]\n swap; \u00b7 rwa [hZZ'x] at hz\n congr\n have : \u03bc.map (\u27e8X, Z\u27e9) (s \u00d7\u02e2 {x.1}) = \u03bc''.map (\u27e8X', Z'\u27e9) (s \u00d7\u02e2 {x.1}) := by rw [h1.map_eq]\n rwa [Measure.map_apply (hX.prod_mk hZ) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hX'.prod_mk hZ') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 X' a) \u207b\u00b9' s)] at this\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this\n\nlemma condRuzsaDist'_of_copy (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} (hY : Measurable Y)\n {W : \u03a9' \u2192 T} (hW : Measurable W)\n (X' : \u03a9'' \u2192 G) {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib X X' \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange W] [FiniteRange W'] :\n d[X ; \u03bc # Y | W ; \u03bc'] = d[X' ; \u03bc'' # Y' | W' ; \u03bc'''] := by\n classical\n set A := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (dirac ()) (\u03bc'.map W)\n ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 A : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n simp only [A]\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n have hfull' : Measure.prod (dirac ()) (\u03bc'''.map W')\n ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 A : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n simp only [A]\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, condRuzsaDist'_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hWW',\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 rw [kernel.const_apply, kernel.const_apply, h1.map_eq]\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this\n\nlemma condRuszaDist_prod_eq_of_indepFun {\u03bc : Measure \u03a9} {\u03bc' : Measure \u03a9'} {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G}\n {W W' : \u03a9' \u2192 T} (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (hW' : Measurable W') (h : IndepFun (\u27e8Y, W\u27e9) W' \u03bc')\n [IsProbabilityMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | \u27e8W, W'\u27e9 ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist'_prod_eq_sum' _ _ hY hW hW']\n have : d[X ; \u03bc # Y | W ; \u03bc'] = \u2211 z, (\u03bc' (W' \u207b\u00b9' {z})).toReal * d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u03bc' hW', one_mul]\n rw [this]\n congr with w\n rcases eq_or_ne (\u03bc' (W' \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n congr 1\n apply condRuzsaDist'_of_copy _ hY hW _ hY hW (IdentDistrib.refl hX.aemeasurable)\n exact (h.identDistrib_cond (MeasurableSet.singleton w) (hY.prod_mk hW) hW' hw).symm\n\nvariable (\u03bc \u03bc') in\nlemma condRuzsaDist_comp_right {T' : Type*} [Fintype T] [Fintype T'] [MeasurableSpace T']\n [MeasurableSingletonClass T'] [IsFiniteMeasure \u03bc']\n (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (e : T \u2192 T')\n (hY : Measurable Y) (hW : Measurable W) (he : Measurable e)\n (h'e : Injective e) :\n d[X ; \u03bc # Y | e \u2218 W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist'_eq_sum' hY (he.comp hW), condRuzsaDist'_eq_sum' hY hW]\n simp [Set.preimage_comp]\n have A i : e \u207b\u00b9' {e i} = {i} := by ext x; simp [h'e.eq_iff]\n symm\n refine Fintype.sum_of_injective e h'e _ _ (fun i hi \u21a6 ?_) (by simp [A])\n suffices e \u207b\u00b9' {i} = \u2205 by simp [this]\n simpa [Set.eq_empty_iff_forall_not_mem] using hi\n\nlemma condRuzsaDist_of_inj_map {G' : Type*} [Countable G'] [AddCommGroup G']\n [MeasurableSpace G'] [MeasurableSingletonClass G'] [IsProbabilityMeasure \u03bc]\n (Y : Fin 4 \u2192 \u03a9 \u2192 G) (h_indep : IndepFun (\u27e8Y 0, Y 2\u27e9) (\u27e8Y 1, Y 3\u27e9) \u03bc)\n (h_meas : \u2200 i, Measurable (Y i)) (\u03c0 : G \u00d7 G \u2192+ G')\n (h\u03c0 : \u2200 (h : G), Function.Injective (fun g \u21a6 \u03c0 (g, h)))\n [FiniteRange (Y 2)] [FiniteRange (Y 3)] :\n d[\u03c0 \u2218 \u27e8Y 0, Y 2\u27e9 | Y 2 ; \u03bc # \u03c0 \u2218 \u27e8Y 1, Y 3\u27e9 | Y 3 ; \u03bc] = d[Y 0 | Y 2 ; \u03bc # Y 1 | Y 3 ; \u03bc] := by\n let f (h : G) (g : G) : G' := \u03c0 (g, h)\n let f' : G \u00d7 G \u2192 G \u2192 G' := fun (h1, h2) \u21a6 fun g \u21a6 \u03c0 (g, h1 - h2)\n have hf' (t : G \u00d7 G) : Function.Injective (f' t) := fun _ _ h \u21a6 h\u03c0 _ h\n let f'' : G \u00d7 G \u2192 G' \u00d7 G := fun (g, h) \u21a6 (\u03c0 (g, h), h)\n have hf'' : Measurable f'' := measurable_of_countable _\n have hm1 : Measurable (Y 0 - Y 1) := (h_meas 0).sub (h_meas 1)\n have hm2 : Measurable (\u27e8Y 2, Y 3\u27e9) := (h_meas 2).prod_mk (h_meas 3)\n rw [condRuzsaDist_of_indep (h_meas 0) (h_meas 2) (h_meas 1) (h_meas 3) \u03bc h_indep,\n condRuzsaDist_of_indep ((measurable_of_countable _).comp ((h_meas 0).prod_mk (h_meas 2)))\n (h_meas 2) ((measurable_of_countable _).comp ((h_meas 1).prod_mk (h_meas 3))) (h_meas 3) \u03bc\n (h_indep.comp hf'' hf''),\n \u2190 condEntropy_of_injective \u03bc hm1 hm2 f' hf', \u2190 map_comp_sub \u03c0,\n \u2190 condEntropy_of_injective \u03bc (h_meas 0) (h_meas 2) f h\u03c0,\n \u2190 condEntropy_of_injective \u03bc (h_meas 1) (h_meas 3) f h\u03c0]\n rfl\n\nlemma condRuzsaDist'_of_inj_map [IsProbabilityMeasure \u03bc] [elem: ElementaryAddCommGroup G 2]\n {X B C : \u03a9 \u2192 G}\n (hX : Measurable X) (hB : Measurable B) (hC : Measurable C)\n (h_indep : IndepFun X (\u27e8B, C\u27e9) \u03bc) [FiniteRange X] [FiniteRange B] [FiniteRange C] :\n d[X ; \u03bc # B | B + C ; \u03bc] = d[X ; \u03bc # C | B + C ; \u03bc] := by\n let \u03c0 : G \u00d7 G \u2192+ G :=\n { toFun := fun x \u21a6 x.2 - x.1\n map_zero' := by simp\n map_add' := fun a b \u21a6 by simp only [Prod.snd_add, Prod.fst_add,\n ElementaryAddCommGroup.sub_eq_add]; abel }\n let Y : Fin 4 \u2192 \u03a9 \u2192 G := ![-X, C, fun _ \u21a6 0, B + C]\n have _ : FiniteRange (Y 0) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 1) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 2) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 3) := by simp [Y]; infer_instance\n\n have hY_meas : \u2200 i, Measurable (Y i) := by\n intro i\n fin_cases i\n exacts [hX.neg, hC, measurable_const, hB.add hC]\n calc d[X ; \u03bc # B | B + C ; \u03bc]\n = d[X | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # B | B + C ; \u03bc] := by\n rw [condRuzsaDist_of_const hX _ _]\n _ = d[\u03c0 \u2218 \u27e8-X, fun _ : \u03a9 \u21a6 (0 : G)\u27e9 | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # \u03c0 \u2218 \u27e8C, B + C\u27e9 | B + C ; \u03bc] := by\n congr\n \u00b7 ext1 \u03c9; simp [\u03c0]\n \u00b7 ext1 \u03c9\n simp only [AddMonoidHom.coe_mk, ZeroHom.coe_mk, Function.comp_apply, Pi.add_apply, \u03c0]\n abel\n _ = d[\u03c0 \u2218 \u27e8Y 0, Y 2\u27e9 | Y 2 ; \u03bc # \u03c0 \u2218 \u27e8Y 1, Y 3\u27e9 | Y 3 ; \u03bc] := by congr\n _ = d[-X | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # C | B + C ; \u03bc] := by\n rw [condRuzsaDist_of_inj_map _ _ hY_meas \u03c0 (fun _ \u21a6 sub_right_injective)]\n \u00b7 congr\n \u00b7 have h1 : (\u27e8Y 0, Y 2\u27e9) = (fun x \u21a6 (-x, 0)) \u2218 X := by ext1 \u03c9; simp [Y]\n have h2 : (\u27e8Y 1, Y 3\u27e9) = (fun p \u21a6 (p.2, p.1 + p.2)) \u2218 (\u27e8B, C\u27e9) := by\n ext1 \u03c9;\n simp only [ElementaryAddCommGroup.neg_eq_self, Matrix.cons_val_one, Matrix.head_cons,\n Function.comp_apply, Prod.mk.injEq, Matrix.cons_val', Pi.add_apply, Matrix.empty_val',\n Matrix.cons_val_fin_one, true_and, Y]\n congr\n rw [h1, h2]\n refine h_indep.comp ?_ ?_\n \u00b7 exact measurable_neg.prod_mk measurable_const\n \u00b7 exact measurable_snd.prod_mk (measurable_fst.add measurable_snd)\n _ = d[-X ; \u03bc # C | B + C ; \u03bc] := by rw [condRuzsaDist_of_const]; exact hX.neg\n _ = d[X ; \u03bc # C | B + C ; \u03bc] := by -- because ElementaryAddCommGroup G 2\n congr\n simp\n\nlemma condRuzsaDist'_of_inj_map' [elem: ElementaryAddCommGroup G 2] [IsProbabilityMeasure \u03bc]\n [IsProbabilityMeasure \u03bc''] {A : \u03a9'' \u2192 G} {B C : \u03a9 \u2192 G} (hA : Measurable A) (hB : Measurable B)\n (hC : Measurable C) [FiniteRange A] [FiniteRange B] [FiniteRange C] :\n d[A ; \u03bc'' # B | B + C ; \u03bc] = d[A ; \u03bc'' # C | B + C ; \u03bc] := by\n -- we want to apply `condRuzsaDist'_of_inj_map'`, but for that all variables need to be in the same\n -- probability space\n let \u03a9' := \u03a9'' \u00d7 \u03a9\n set X\u2082' : \u03a9' \u2192 G := A \u2218 Prod.fst with hX\u2082'_def\n have hX\u2082' : Measurable X\u2082' := hA.comp measurable_fst\n let B' : \u03a9' \u2192 G := B \u2218 Prod.snd\n have hB' : Measurable B' := hB.comp measurable_snd\n let C' : \u03a9' \u2192 G := C \u2218 Prod.snd\n have hC' : Measurable C' := hC.comp measurable_snd\n let \u03bc' : Measure \u03a9' := Measure.prod \u03bc'' \u03bc\n haveI : IsProbabilityMeasure \u03bc' := by infer_instance\n -- h1 and h2 should be applications of a new lemma?\n have h1 : d[A ; \u03bc'' # B | B + C ; \u03bc] = d[X\u2082' ; \u03bc' # B' | B' + C' ; \u03bc'] := by\n refine condRuzsaDist'_of_copy A hB (by measurability) X\u2082' hB' (by measurability) ?_ ?_\n \u00b7 constructor\n \u00b7 exact hA.aemeasurable\n \u00b7 exact hX\u2082'.aemeasurable\n \u00b7 rw [\u2190 Measure.map_map hA measurable_fst]\n simp [\u03bc']\n \u00b7 constructor\n \u00b7 exact (hB.prod_mk (hB.add hC)).aemeasurable\n \u00b7 exact (hB'.prod_mk (hB'.add hC')).aemeasurable\n \u00b7 have : \u27e8B', B' + C'\u27e9 = (\u27e8B, B + C\u27e9) \u2218 Prod.snd := by ext1 _; simp [C', B']\n rw [this, \u2190 Measure.map_map _ measurable_snd]\n \u00b7 simp only [Measure.map_snd_prod, measure_univ, one_smul, \u03bc']\n \u00b7 exact hB.prod_mk (hB.add hC)\n have h2 : d[A ; \u03bc'' # C | B + C ; \u03bc] = d[X\u2082' ; \u03bc' # C' | B' + C' ; \u03bc'] := by\n apply condRuzsaDist'_of_copy _ hC (by measurability) X\u2082' hC' (by measurability) ?_ ?_\n \u00b7 constructor\n \u00b7 exact hA.aemeasurable\n \u00b7 exact hX\u2082'.aemeasurable\n \u00b7 rw [\u2190 Measure.map_map hA measurable_fst]\n simp [\u03bc']\n \u00b7 constructor\n \u00b7 exact (hC.prod_mk (hB.add hC)).aemeasurable\n \u00b7 exact (hC'.prod_mk (hB'.add hC')).aemeasurable\n \u00b7 have : \u27e8C', B' + C'\u27e9 = (\u27e8C, B + C\u27e9) \u2218 Prod.snd := by ext1 _; simp [B', C']\n rw [this, \u2190 Measure.map_map _ measurable_snd]\n \u00b7 simp only [Measure.map_snd_prod, measure_univ, one_smul, \u03bc']\n \u00b7 exact hC.prod_mk (hB.add hC)\n rw [h1, h2, condRuzsaDist'_of_inj_map hX\u2082' hB' hC']\n rw [indepFun_iff_map_prod_eq_prod_map_map hX\u2082'.aemeasurable (hB'.prod_mk hC').aemeasurable]\n have h_prod : (fun \u03c9 \u21a6 (X\u2082' \u03c9, prod B' C' \u03c9)) = Prod.map A (\u27e8B, C\u27e9) := by ext1; simp [B', C', X\u2082']\n have h_comp_snd : (fun a \u21a6 (B' a, C' a)) = (\u27e8B, C\u27e9) \u2218 Prod.snd := by ext1; simp [B', C']\n rw [h_prod, h_comp_snd, hX\u2082'_def, \u2190 Measure.map_map _ measurable_snd,\n \u2190 Measure.map_map _ measurable_fst, Measure.map_prod_map]\n rotate_left\n \u00b7 exact hA\n \u00b7 exact hB.prod_mk hC\n \u00b7 exact hA\n \u00b7 exact hB.prod_mk hC\n simp [\u03bc']\n\n/-- The **Kaimanovich-Vershik inequality**. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$ -/\nlemma kaimanovich_vershik {X Y Z : \u03a9 \u2192 G} (h : iIndepFun (fun _ \u21a6 hG) ![X, Y, Z] \u03bc)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [IsProbabilityMeasure \u03bc]\n [FiniteRange X] [FiniteRange Z] [FiniteRange Y] :\n H[X + Y + Z ; \u03bc] - H[X + Y ; \u03bc] \u2264 H[Y + Z ; \u03bc] - H[Y ; \u03bc] := by\n suffices (H[X ; \u03bc] + H[Y ; \u03bc] + H[Z ; \u03bc]) + H[X + Y + Z ; \u03bc]\n \u2264 (H[X ; \u03bc] + H[Y + Z ; \u03bc]) + (H[Z ; \u03bc] + H[X + Y ; \u03bc]) by linarith\n have : \u2200 (i : Fin 3), Measurable (![X, Y, Z] i) := fun i \u21a6 by fin_cases i <;> assumption\n convert entropy_triple_add_entropy_le _ hX hZ (show Measurable (X + (Y + Z)) by measurability)\n using 2\n . calc\n H[X ; \u03bc] + H[Y ; \u03bc] + H[Z ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [IndepFun.entropy_pair_eq_add hX hY]\n convert h.indepFun (show 0 \u2260 1 by decide)\n _ = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n rw [IndepFun.entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h.indepFun_prod_mk this 0 1 2 (by decide) (by decide)\n _ = H[\u27e8X, \u27e8Z , X + (Y + Z)\u27e9\u27e9 ; \u03bc] := by\n apply entropy_of_comp_eq_of_comp \u03bc (by measurability) (by measurability)\n (fun ((x, y), z) \u21a6 (x, z, x + y + z)) (fun (a, b, c) \u21a6 ((a, c - a - b), b))\n all_goals { funext \u03c9; dsimp [prod]; ext <;> dsimp; abel }\n . rw [add_assoc]\n . symm\n refine (entropy_add_right hX (by measurability) _).trans $\n IndepFun.entropy_pair_eq_add hX (by measurability) ?_\n exact h.indepFun_add_right this 0 1 2 (by decide) (by decide)\n \u00b7 rw [eq_comm, \u2190 add_assoc]\n refine (entropy_add_right' hZ (by measurability) _).trans $\n IndepFun.entropy_pair_eq_add hZ (by measurability) ?_\n exact h.indepFun_add_right this 2 0 1 (by decide) (by decide)\n\n/-- A version of the **Kaimanovich-Vershik inequality** with some variables negated. -/\nlemma kaimanovich_vershik' {X Y Z : \u03a9 \u2192 G} (h : iIndepFun (fun _ \u21a6 hG) ![X, Y, Z] \u03bc)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [IsProbabilityMeasure \u03bc]\n [FiniteRange X] [FiniteRange Z] [FiniteRange Y] :\n H[X - (Y + Z) ; \u03bc] - H[X - Y ; \u03bc] \u2264 H[Y + Z ; \u03bc] - H[Y ; \u03bc] := by\n rw [\u2190 entropy_neg (hY.add' hZ), \u2190 entropy_neg hY]\n simp_rw [sub_eq_add_neg, neg_add, \u2190 add_assoc]\n refine kaimanovich_vershik ?_ hX ?_ hZ.neg\n convert (h.neg 1).neg 2\n ext i; fin_cases i\n \u00b7 simp (discharger := decide)\n \u00b7 simp (discharger := decide)\n \u00b7 rw [\u2190 show \u2200 h : 2 < 3, (2 : Fin 3) = \u27e82, h\u27e9 by intro; rfl]\n simp (discharger := decide)\n . exact hY.neg\n\nsection BalogSzemerediGowers\n\n/-- The **entropic Balog-Szemer\u00e9di-Gowers inequality**. Let $A, B$ be $G$-valued random variables on\n$\\Omega$, and set $Z := A+B$. Then\n$$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]. $$\nTODO: remove the hypothesis of `Fintype G` from here and from `condIndep_copies'` -/\nlemma ent_bsg [IsProbabilityMeasure \u03bc] {A B : \u03a9 \u2192 G} (hA : Measurable A) (hB : Measurable B)\n [Fintype G] :\n (\u03bc.map (A + B))[fun z \u21a6 d[A ; \u03bc[|(A + B) \u207b\u00b9' {z}] # B ; \u03bc[|(A + B) \u207b\u00b9' {z}]]]\n \u2264 3 * I[A : B; \u03bc] + 2 * H[A + B ; \u03bc] - H[A ; \u03bc] - H[B ; \u03bc] := by\n let Z := A + B\n have hZ : Measurable Z := hA.add hB\n obtain \u27e8\u03a9', _, AB\u2081, AB\u2082, Z', \u03bd, _, hAB\u2081, hAB\u2082, hZ', hABZ, hABZ\u2081, hABZ\u2082, hZ\u2081, hZ\u2082\u27e9 :=\n condIndep_copies' (\u27e8A, B\u27e9) Z (hA.prod_mk hB) hZ \u03bc (fun (a, b) c \u21a6 c = a + b)\n (measurable_discrete _) (eventually_of_forall fun _ \u21a6 rfl)\n let A\u2081 := fun \u03c9 \u21a6 (AB\u2081 \u03c9).1\n let B\u2081 := fun \u03c9 \u21a6 (AB\u2081 \u03c9).2\n let A\u2082 := fun \u03c9 \u21a6 (AB\u2082 \u03c9).1\n let B\u2082 := fun \u03c9 \u21a6 (AB\u2082 \u03c9).2\n replace hZ\u2081 : Z' = A\u2081 + B\u2081 := funext hZ\u2081\n replace hZ\u2082 : Z' = A\u2082 + B\u2082 := funext hZ\u2082\n have hA\u2081 : Measurable A\u2081 := hAB\u2081.fst\n have hB\u2081 : Measurable B\u2081 := hAB\u2081.snd\n have hA\u2082 : Measurable A\u2082 := hAB\u2082.fst\n have hB\u2082 : Measurable B\u2082 := hAB\u2082.snd\n have hZZ' : IdentDistrib Z' Z \u03bd \u03bc := hABZ\u2081.comp measurable_snd\n have :=\n calc\n H[\u27e8A\u2081, \u27e8B\u2081, A\u2081 - B\u2082\u27e9\u27e9 ; \u03bd]\n = H[\u27e8\u27e8A\u2081, B\u2081\u27e9, \u27e8\u27e8A\u2082, B\u2082\u27e9, Z'\u27e9\u27e9 ; \u03bd] := entropy_of_comp_eq_of_comp _\n (hA\u2081.prod_mk $ hB\u2081.prod_mk $ hA\u2081.sub hB\u2082) (hAB\u2081.prod_mk $ hAB\u2082.prod_mk hZ')\n (fun (a, b, c) \u21a6 ((a, b), (b + c, a - c), a + b))\n (fun ((a, b), (_c, d), _e) \u21a6 (a, b, a - d))\n (by funext; simpa [sub_add_eq_add_sub, Prod.ext_iff, \u2190 hZ\u2081, hZ\u2082, two_nsmul, \u2190 add_sub_assoc,\n add_comm, eq_sub_iff_add_eq] using congr_fun (hZ\u2082.symm.trans hZ\u2081) _) rfl\n _ = H[\u27e8\u27e8A\u2081, B\u2081\u27e9, Z'\u27e9 ; \u03bd] + H[\u27e8\u27e8A\u2082, B\u2082\u27e9, Z'\u27e9 ; \u03bd] - H[Z' ; \u03bd] :=\n ent_of_cond_indep hAB\u2081 hAB\u2082 hZ' hABZ\n _ = 2 * H[\u27e8\u27e8A, B\u27e9, Z\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n rw [two_mul]\n congr 1\n congr 1 <;> exact IdentDistrib.entropy_eq \u2039_\u203a\n exact hZZ'.entropy_eq\n _ = 2 * H[\u27e8A, B\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n congr 2\n exact entropy_prod_comp (hA.prod_mk hB) _ fun x \u21a6 x.1 + x.2\n have :=\n calc\n H[\u27e8A\u2081, A\u2081 - B\u2082\u27e9 ; \u03bd]\n = H[\u27e8A\u2081, B\u2082\u27e9 ; \u03bd] := entropy_sub_right hA\u2081 hB\u2082 _\n _ \u2264 H[A\u2081 ; \u03bd] + H[B\u2082 ; \u03bd] := entropy_pair_le_add hA\u2081 hB\u2082 _\n _ = H[A ; \u03bc] + H[B ; \u03bc] := by\n congr 1\n exact (hABZ\u2081.comp measurable_fst.fst).entropy_eq\n exact (hABZ\u2082.comp measurable_fst.snd).entropy_eq\n have :=\n calc\n H[\u27e8B\u2081, A\u2081 - B\u2082\u27e9 ; \u03bd]\n = H[\u27e8A\u2082, B\u2081\u27e9 ; \u03bd] := by\n rw [entropy_comm hB\u2081 (show Measurable (A\u2081 - B\u2082) from hA\u2081.sub hB\u2082),\n \u2190 entropy_sub_left' hA\u2082 hB\u2081, sub_eq_sub_iff_add_eq_add.2 $ hZ\u2081.symm.trans hZ\u2082]\n _ \u2264 H[A\u2082 ; \u03bd] + H[B\u2081 ; \u03bd] := entropy_pair_le_add hA\u2082 hB\u2081 _\n _ = H[A ; \u03bc] + H[B ; \u03bc] := by\n congr 1\n exact (hABZ\u2082.comp measurable_fst.fst).entropy_eq\n exact (hABZ\u2081.comp measurable_fst.snd).entropy_eq\n have :=\n calc\n _ \u2264 _ := entropy_triple_add_entropy_le \u03bd hA\u2081 hB\u2081 (show Measurable (A\u2081 - B\u2082) from hA\u2081.sub hB\u2082)\n _ \u2264 _ := add_le_add \u2039_\u203a \u2039_\u203a\n have :=\n calc\n H[A\u2081 - B\u2082 | Z' ; \u03bd]\n \u2264 H[A\u2081 - B\u2082 ; \u03bd] := condEntropy_le_entropy _ (hA\u2081.sub hB\u2082) hZ'\n _ \u2264 _ := le_sub_iff_add_le'.2 \u2039_\u203a\n _ = 2 * I[A : B ; \u03bc] + H[Z ; \u03bc] := by\n rw [\u2039H[\u27e8A\u2081, \u27e8B\u2081, A\u2081 - B\u2082\u27e9\u27e9 ; \u03bd] = _\u203a, mutualInfo_def]; ring\n have hA\u2081Z :=\n calc\n H[A\u2081 | Z' ; \u03bd]\n _ = H[\u27e8A\u2081, B\u2081\u27e9 ; \u03bd] - H[Z' ; \u03bd] := by\n rw [chain_rule'', hZ\u2081, entropy_add_right, entropy_comm] <;> assumption\n _ = H[\u27e8A, B\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n congr 1\n exact (hABZ\u2081.comp measurable_fst).entropy_eq\n exact hZZ'.entropy_eq\n _ = H[A ; \u03bc] + H[B ; \u03bc] - I[A : B ; \u03bc] - H[Z ; \u03bc] := by\n rw [\u2190 entropy_add_entropy_sub_mutualInfo]\n have hB\u2082Z :=\n calc\n H[B\u2082 | Z' ; \u03bd]\n _ = H[\u27e8A\u2082, B\u2082\u27e9 ; \u03bd] - H[Z' ; \u03bd] := by\n rw [chain_rule'', hZ\u2082, entropy_add_right', entropy_comm] <;> assumption\n _ = H[\u27e8A, B\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n congr 1\n exact (hABZ\u2082.comp measurable_fst).entropy_eq\n exact hZZ'.entropy_eq\n _ = H[A ; \u03bc] + H[B ; \u03bc] - I[A : B ; \u03bc] - H[Z ; \u03bc] := by\n rw [\u2190 entropy_add_entropy_sub_mutualInfo]\n save\n calc\n (\u03bc.map Z)[fun z \u21a6 d[A ; \u03bc[|Z \u2190 z] # B ; \u03bc[|Z \u2190 z]]]\n = (\u03bd.map Z')[fun z \u21a6 d[A\u2081 ; \u03bd[|Z' \u2190 z] # B\u2082 ; \u03bd[|Z' \u2190 z]]] := by\n rw [hZZ'.map_eq]\n refine' integral_congr_ae $ eventually_of_forall fun z \u21a6 _\n have hAA\u2081 : IdentDistrib A\u2081 A (\u03bd[|Z' \u2190 z]) (\u03bc[|Z \u2190 z]) :=\n (hABZ\u2081.comp $ measurable_fst.fst.prod_mk measurable_snd).cond\n (measurableSet_singleton z) hZ' hZ\n have hBB\u2082 : IdentDistrib B\u2082 B (\u03bd[|Z' \u2190 z]) (\u03bc[|Z \u2190 z]) :=\n (hABZ\u2082.comp $ measurable_fst.snd.prod_mk measurable_snd).cond\n (measurableSet_discrete _) hZ' hZ\n dsimp (config := {zeta := false}) [rdist]\n rw [\u2190 hAA\u2081.entropy_eq, \u2190 hBB\u2082.entropy_eq, hAA\u2081.map_eq, hBB\u2082.map_eq]\n _ = (\u03bd.map Z')[fun z \u21a6\n H[A\u2081 - B\u2082 ; \u03bd[|Z' \u2190 z]] - H[A\u2081 ; \u03bd[|Z' \u2190 z]]/2 - H[B\u2082 ; \u03bd[|Z' \u2190 z]]/2] :=\n integral_congr_ae $ hABZ.mono fun z hz \u21a6\n (hz.comp measurable_fst measurable_snd).rdist_eq hA\u2081 hB\u2082\n _ = H[A\u2081 - B\u2082 | Z' ; \u03bd] - H[A\u2081 | Z' ; \u03bd] / 2 - H[B\u2082 | Z' ; \u03bd] / 2 := by\n rw [integral_sub, integral_sub, integral_div, integral_div]\n rfl\n all_goals exact .of_finite _ _\n _ \u2264 2 * I[A : B ; \u03bc] + H[Z ; \u03bc] - H[A\u2081 | Z' ; \u03bd] / 2 - H[B\u2082 | Z' ; \u03bd] / 2 :=\n sub_le_sub_right (sub_le_sub_right \u2039_\u203a _) _\n _ = _ := by rw [hA\u2081Z, hB\u2082Z]; ring\n\nend BalogSzemerediGowers\n\nvariable (\u03bc \u03bc') in\n", "theoremStatement": "/-- Suppose that $(X, Z)$ and $(Y, W)$ are random variables, where $X, Y$ take values in an abelian\ngroup. Then $$d[X | Z ; Y | W] \\leq d[X ; Y] + \\tfrac{1}{2} I[X : Z] + \\tfrac{1}{2} I[Y : W]$$ -/\nlemma condRuzsaDist_le {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [Nonempty S]\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n [FiniteRange X] [FiniteRange Z] [FiniteRange Y] [FiniteRange W] :\n d[X | Z ; \u03bc # Y|W ; \u03bc'] \u2264 d[X ; \u03bc # Y ; \u03bc'] + I[X : Z ; \u03bc]/2 + I[Y : W ; \u03bc']/2 ", "theoremName": "condRuzsaDist_le", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fdcd5fb746226cf80f93291191e0379295b", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 1238, "tokenPositionInFile": 62312, 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"Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have hXZ : Measurable (\u27e8X, Z\u27e9 : \u03a9 \u2192 G \u00d7 S):= Measurable.prod_mk hX hZ\n have hYW : Measurable (\u27e8Y, W\u27e9 : \u03a9' \u2192 G \u00d7 T):= Measurable.prod_mk hY hW\n obtain \u27e8\u03bd, XZ', YW', _, hXZ', hYW', hind, hIdXZ, hIdYW, _, _\u27e9 :=\n independent_copies_finiteRange hXZ hYW \u03bc \u03bc'\n let X' := Prod.fst \u2218 XZ'\n let Z' := Prod.snd \u2218 XZ'\n let Y' := Prod.fst \u2218 YW'\n let W' := Prod.snd \u2218 YW'\n have hX' : Measurable X' := hXZ'.fst\n have hZ' : Measurable Z' := hXZ'.snd\n have hY' : Measurable Y' := hYW'.fst\n have hW' : Measurable W' := hYW'.snd\n have hind' : IndepFun X' Y' \u03bd := hind.comp measurable_fst measurable_fst\n rw [show XZ' = \u27e8X', Z'\u27e9 by rfl] at hIdXZ hind\n rw [show YW' = \u27e8Y', W'\u27e9 by rfl] at hIdYW hind\n rw [\u2190 condRuzsaDist_of_copy hX' hZ' hY' hW' hX hZ hY hW hIdXZ hIdYW,\n condRuzsaDist_of_indep hX' hZ' hY' hW' _ hind]\n have hIdX : IdentDistrib X X' \u03bc \u03bd := hIdXZ.symm.comp measurable_fst\n have hIdY : IdentDistrib Y Y' \u03bc' \u03bd := hIdYW.symm.comp measurable_fst\n rw [hIdX.rdist_eq hIdY, hIdXZ.symm.mutualInfo_eq, hIdYW.symm.mutualInfo_eq,\n hind'.rdist_eq hX' hY', mutualInfo_eq_entropy_sub_condEntropy hX' hZ',\n mutualInfo_eq_entropy_sub_condEntropy hY' hW']\n have h := condEntropy_le_entropy \u03bd (X := X' - Y') (hX'.sub hY') (hZ'.prod_mk hW')\n linarith [h, entropy_nonneg Z' \u03bd, entropy_nonneg W' \u03bd]", "proofType": "tactic", "proofLengthLines": 24, "proofLengthTokens": 1308}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\nlemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal := by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]\n\nlemma entropy_eq_sum_finset' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finset hX hA\n\nlemma entropy_eq_sum_finiteRange (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog (\u03bc.map X {x}).toReal := entropy_eq_sum_finset hX (A := FiniteRange.toFinset X) (full_measure_of_finiteRange hX)\n\nlemma entropy_eq_sum_finiteRange' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finiteRange hX\n\n/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\nlemma entropy_cond_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) :\n H[X | Y \u2190 y ; \u03bc] = \u2211' x, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum hX]\n\nlemma entropy_cond_eq_sum_finiteRange (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) [FiniteRange X]:\n H[X | Y \u2190 y ; \u03bc] = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum_finiteRange hX]\n\n/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\nlemma entropy_comp_of_injective\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 T) (hf : Function.Injective f) :\n H[f \u2218 X ; \u03bc] = H[X ; \u03bc] := by\n have hf_m : Measurable f := measurable_of_countable f\n rw [entropy_def, \u2190 Measure.map_map hf_m hX, measureEntropy_map_of_injective _ _ hf_m hf,\n entropy_def]\n\n\n/-- The entropy of any constant is zero. -/\n@[simp] lemma entropy_const [IsProbabilityMeasure \u03bc] (c : S) : H[fun _ \u21a6 c ; \u03bc] = 0 := by\n simp [entropy, Measure.map_const]\n\nopen Set\n\nopen Function\n\n/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\nlemma IsUniform.entropy_eq (H : Finset S) (X : \u03a9 \u2192 S) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n haveI : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX'.aemeasurable\n have (t : S) : negMulLog ((\u03bc.map X).real {t}) = ((\u03bc.map X).real {t}) * log (Nat.card H) := by\n by_cases ht : t \u2208 H\n \u00b7 simp only [negMulLog, neg_mul, neg_mul_eq_mul_neg, IsUniform.measureReal_preimage_of_mem'\n hX hX' ht, one_div, log_inv, neg_neg]\n \u00b7 simp only [negMulLog, map_measureReal_apply hX' (MeasurableSet.singleton t),\n IsUniform.measureReal_preimage_of_nmem hX ht, neg_zero, log_zero, mul_zero, zero_mul]\n rw [entropy_eq_sum_finset' (A := H) hX', Finset.sum_congr rfl (fun t _ \u21a6 this t), \u2190 Finset.sum_mul,\n Finset.sum_realMeasure_singleton]\n . convert one_mul _\n have := IsUniform.full_measure hX hX'\n simp at this\n rw [Measure.real, this]\n simp\n rw [Measure.map_apply hX' (by measurability)]\n exact hX.measure_preimage_compl\n\n\n/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\nlemma IsUniform.entropy_eq' {H : Set S} [Finite H] {X : \u03a9 \u2192 S} {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n convert IsUniform.entropy_eq H.toFinite.toFinset X ?_ hX' with x\n . simp; exact Iff.rfl\n . infer_instance\n convert hX\n simp\n\n/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\nlemma entropy_eq_log_card {X : \u03a9 \u2192 S} [Fintype S] (hX : Measurable X) (\u03bc : Measure \u03a9) [h\u03bc : NeZero \u03bc]\n [IsFiniteMeasure \u03bc] :\n entropy X \u03bc = log (Fintype.card S) \u2194 \u2200 s, \u03bc.map X {s} = \u03bc Set.univ / Fintype.card S := by\n rcases eq_zero_or_neZero (\u03bc.map X) with h | h\n . have := Measure.le_map_apply (@Measurable.aemeasurable \u03a9 S _ _ X \u03bc hX) Set.univ\n simp [h] at this; simp [this] at h\u03bc\n have : IsFiniteMeasure (\u03bc.map X) := by\n apply Measure.isFiniteMeasure_map\n rw [entropy_def, measureEntropy_eq_card_iff_measure_eq, Measure.map_apply hX MeasurableSet.univ]\n simp\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\nlemma prob_ge_exp_neg_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) [hX': FiniteRange X] :\n \u2203 s : S, \u03bc.map X {s} \u2265 (\u03bc Set.univ) * (rexp (- H[X ; \u03bc])).toNNReal := by\n let \u03bcS := \u03bc.map X\n let \u03bcs s := \u03bcS {s}\n rcases finiteSupport_of_finiteRange (X := X) with \u27e8A, hA\u27e9\n\n let S_nonzero := A.filter (fun s \u21a6 \u03bcs s \u2260 0)\n\n set norm := \u03bcS A with rw_norm\n have h_norm : norm = \u03bc Set.univ := by\n have := measure_add_measure_compl (\u03bc := \u03bcS) (s := A) (Finset.measurableSet _)\n rw [hA, add_zero] at this\n simp [norm, \u03bcS, this, Measure.map_apply hX MeasurableSet.univ]\n\n let pdf_nn s := norm\u207b\u00b9 * \u03bcs s\n let pdf s := (pdf_nn s).toReal\n let neg_log_pdf s := -log (pdf s)\n\n rcases Finset.eq_empty_or_nonempty S_nonzero with h_empty | h_nonempty\n . have h_norm_zero : \u03bc Set.univ = 0 := by\n have h : \u2200 s \u2208 A, \u03bcs s \u2260 0 \u2192 \u03bcs s \u2260 0 := fun _ _ h \u21a6 h\n rw [\u2190 h_norm, rw_norm, \u2190 Finset.sum_measure_singleton, \u2190 Finset.sum_filter_of_ne h,\n show Finset.filter _ _ = S_nonzero from rfl, h_empty, show Finset.sum \u2205 \u03bcs = 0 from rfl]\n use Classical.arbitrary (\u03b1 := S)\n rw [h_norm_zero, zero_mul]\n exact le_of_not_gt ENNReal.not_lt_zero\n\n rcases exists_or_forall_not (fun s \u21a6 \u03bc.map X {s} = \u221e) with h_infty | h_finite\n . obtain \u27e8s, h_s\u27e9 := h_infty\n use s; rw [h_s] ; exact le_top\n\n rcases eq_zero_or_neZero \u03bc with h_zero_measure | _\n . use Classical.arbitrary (\u03b1 := S)\n rw [h_zero_measure, show (0 : Measure \u03a9) _ = 0 from rfl, zero_mul]\n exact zero_le _\n\n have h_norm_pos : 0 < norm := by\n rw [h_norm, Measure.measure_univ_pos]\n exact NeZero.ne \u03bc\n have h_norm_finite : norm < \u221e := by\n rw [rw_norm, \u2190 Finset.sum_measure_singleton]\n exact ENNReal.sum_lt_top (fun s _ \u21a6 h_finite s)\n have h_invinvnorm_finite : norm\u207b\u00b9\u207b\u00b9 \u2260 \u221e := by\n rw [inv_inv]\n exact LT.lt.ne_top h_norm_finite\n have h_invnorm_ne_zero : norm\u207b\u00b9 \u2260 0 := ENNReal.inv_ne_top.mp h_invinvnorm_finite\n have h_invnorm_finite : norm\u207b\u00b9 \u2260 \u221e := by\n rw [\u2190 ENNReal.inv_ne_zero, inv_inv]\n exact ne_zero_of_lt h_norm_pos\n have h_pdf_finite : \u2200 s, pdf_nn s \u2260 \u221e := fun s \u21a6 ENNReal.mul_ne_top h_invnorm_finite (h_finite s)\n\n have h_norm_cancel : norm * norm\u207b\u00b9 = 1 :=\n ENNReal.mul_inv_cancel (ne_zero_of_lt h_norm_pos) (LT.lt.ne_top h_norm_finite)\n have h_pdf1 : (\u2211 s in A, pdf s) = 1 := by\n rw [\u2190 ENNReal.toReal_sum (fun s _ \u21a6 h_pdf_finite s), \u2190 Finset.mul_sum,\n Finset.sum_measure_singleton, mul_comm, h_norm_cancel, ENNReal.one_toReal]\n\n let \u27e8s_max, hs, h_min\u27e9 := Finset.exists_min_image S_nonzero neg_log_pdf h_nonempty\n have h_pdf_s_max_pos : 0 < pdf s_max := by\n rw [Finset.mem_filter] at hs\n have h_nonzero : pdf s_max \u2260 0 := ENNReal.toReal_ne_zero.mpr\n \u27e8mul_ne_zero h_invnorm_ne_zero hs.2, ENNReal.mul_ne_top h_invnorm_finite (h_finite s_max)\u27e9\n exact LE.le.lt_of_ne ENNReal.toReal_nonneg h_nonzero.symm\n\n use s_max\n rw [\u2190 h_norm, \u2190 one_mul (\u03bc.map X _), \u2190 h_norm_cancel, mul_assoc]\n apply mul_le_mul_of_nonneg_left _ (le_of_lt h_norm_pos)\n show ENNReal.ofReal (rexp (-H[X ; \u03bc])) \u2264 pdf_nn s_max\n rw [ENNReal.ofReal_le_iff_le_toReal (h_pdf_finite _),\n show (pdf_nn _).toReal = pdf _ from rfl, \u2190 Real.exp_log h_pdf_s_max_pos]\n apply exp_monotone\n rw [neg_le, \u2190 one_mul (-log _), \u2190 h_pdf1, Finset.sum_mul]\n let g_lhs s := pdf s * neg_log_pdf s_max\n let g_rhs s := -pdf s * log (pdf s)\n suffices \u2211 s in A, g_lhs s \u2264 \u2211 s in A, g_rhs s by\n convert this\n rw [entropy_eq_sum_finset hX hA]\n congr with s\n simp only [negMulLog, neg_mul, ENNReal.toReal_mul, neg_inj, g_rhs, pdf, pdf_nn]\n simp at h_norm\n rw [h_norm]\n simp\n have h_lhs : \u2200 s, \u03bcs s = 0 \u2192 g_lhs s = 0 := by {intros _ h; simp [g_lhs, pdf, pdf_nn, h]}\n have h_rhs : \u2200 s, \u03bcs s = 0 \u2192 g_rhs s = 0 := by {intros _ h; simp [g_rhs, pdf, pdf_nn, h]}\n rw [\u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_lhs s).mt),\n \u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_rhs s).mt)]\n apply Finset.sum_le_sum\n intros s h_s\n rw [show g_lhs s = _ * _ from rfl, show g_rhs s = _ * _ from rfl, neg_mul_comm]\n exact mul_le_mul_of_nonneg_left (h_min s h_s) ENNReal.toReal_nonneg\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\nlemma prob_ge_exp_neg_entropy' {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (X : \u03a9 \u2192 S) (hX : Measurable X) [FiniteRange X] :\n \u2203 s : S, rexp (- H[X ; \u03bc]) \u2264 \u03bc.real (X \u207b\u00b9' {s}) := by\n obtain \u27e8s, hs\u27e9 := prob_ge_exp_neg_entropy X \u03bc hX\n use s\n rwa [IsProbabilityMeasure.measure_univ, one_mul, ge_iff_le,\n (show ENNReal.ofNNReal _ = ENNReal.ofReal _ from rfl),\n ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _), \u2190 Measure.real,\n map_measureReal_apply hX (MeasurableSet.singleton s)] at hs\n\n/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\nlemma const_of_nonpos_entropy {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] {X : \u03a9 \u2192 S} (hX : Measurable X) [FiniteRange X] (hent: H[X; \u03bc] \u2264 0):\n \u2203 s : S, \u03bc.real (X \u207b\u00b9' {s}) = 1 := by\n rcases prob_ge_exp_neg_entropy' (\u03bc := \u03bc) X hX with \u27e8 s, hs \u27e9\n use s\n apply LE.le.antisymm\n . rw [\u2190 IsProbabilityMeasure.measureReal_univ (\u03bc := \u03bc)]\n exact measureReal_mono (subset_univ _) (by finiteness)\n refine le_trans ?_ hs\n simp [hent]\n\n\n/-- $H[X, Y] = H[Y, X]$. -/\nlemma entropy_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n change H[Prod.swap \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hY.prod_mk hX) Prod.swap Prod.swap_injective\n\n/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\nlemma entropy_assoc (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n change H[MeasurableEquiv.prodAssoc \u2218 \u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc ((hX.prod_mk hY).prod_mk hZ) _ $ Equiv.injective _\n\n/-- $H[X, f(X)] = H[X]$.-/\n@[simp] lemma entropy_prod_comp (hX : Measurable X) (\u03bc : Measure \u03a9) (f : S \u2192 T) :\n H[\u27e8X, f \u2218 X\u27e9; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) fun _ _ ab \u21a6 (Prod.ext_iff.1 ab).1\n\nend entropy\n\nsection condEntropy\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n\n/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\nnoncomputable\ndef condEntropy (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]]\n\nlemma condEntropy_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n condEntropy X Y \u03bc = (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]] := rfl\n\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \" ; \" \u03bc \"]\" => condEntropy X Y \u03bc\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \"]\" => condEntropy X Y volume\n\nlemma condEntropy_eq_zero\n (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (t : T) (ht : (\u03bc.map Y {t}).toReal = 0) : H[X | Y \u2190 t ; \u03bc] = 0 := by\n convert entropy_zero_measure X\n apply cond_eq_zero_of_meas_eq_zero\n rw [Measure.map_apply hY (MeasurableSet.singleton t)] at ht\n rw [\u2190 measureReal_eq_zero_iff]\n exact ht\n\n/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\nlemma condEntropy_eq_kernel_entropy\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] [FiniteRange Y] :\n H[X | Y ; \u03bc] = Hk[condDistrib X Y \u03bc, \u03bc.map Y] := by\n rw [condEntropy_def, kernel.entropy]\n apply integral_congr_finiteSupport\n intro t ht\n simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top (\u03bc.map Y), or_false] at ht\n rw [Measure.map_apply hY (measurableSet_singleton _)] at ht\n simp only [entropy_def]\n congr\n ext S hS\n rw [condDistrib_apply' hX hY _ _ ht hS, Measure.map_apply hX hS,\n cond_apply _ (hY (measurableSet_singleton _))]\n\n/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\nlemma map_prod_comap_swap (hX : Measurable X) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9))).comap Prod.swap = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext s hs\n rw [Measure.map_apply (hZ.prod_mk hX) hs, Measure.comap_apply _ Prod.swap_injective _ _ hs]\n \u00b7 rw [Measure.map_apply (hX.prod_mk hZ)]\n \u00b7 congr with \u03c9\n simp only [Set.image_swap_eq_preimage_swap, Set.mem_preimage, Prod.swap_prod_mk]\n \u00b7 exact MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' hs\n \u00b7 exact fun t ht \u21a6 MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' ht\n\n", "theoremStatement": "lemma condEntropy_two_eq_kernel_entropy (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] =\n Hk[kernel.condKernel (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc),\n Measure.map Z \u03bc \u2297\u2098 kernel.fst (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc)] ", "theoremName": "ProbabilityTheory.condEntropy_two_eq_kernel_entropy", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "0eeeb00408c7957671f8f999b1a27effb8677683", "date": "2023-11-19"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 394, "tokenPositionInFile": 18550, "theoremPositionInFile": 39}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 15, "numPremises": 66, "importedModules": ["Init.Prelude", 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"Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "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.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.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.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", 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"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.Paths", "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", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "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", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "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", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "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", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", 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"Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.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.Lemma", "Mathlib.Tactic.TypeStar", "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.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.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.Init.Data.Int.Basic", "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.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "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.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.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.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "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.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.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [Measure.compProd_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc),\n map_compProd_condDistrib hY hZ,\n kernel.entropy_congr (condKernel_condDistrib_ae_eq hY hX hZ \u03bc),\n \u2190 kernel.entropy_congr (swap_condDistrib_ae_eq hY hX hZ \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, Y \u03c9)) = (\u03bc.map (fun \u03c9 \u21a6 (Y \u03c9, Z \u03c9))).comap Prod.swap := by\n rw [map_prod_comap_swap hY hZ]\n rw [this, condEntropy_eq_kernel_entropy hX (hY.prod_mk hZ), kernel.entropy_comap_swap]", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 457}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\nlemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal := by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]\n\nlemma entropy_eq_sum_finset' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finset hX hA\n\nlemma entropy_eq_sum_finiteRange (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog (\u03bc.map X {x}).toReal := entropy_eq_sum_finset hX (A := FiniteRange.toFinset X) (full_measure_of_finiteRange hX)\n\nlemma entropy_eq_sum_finiteRange' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finiteRange hX\n\n/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\nlemma entropy_cond_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) :\n H[X | Y \u2190 y ; \u03bc] = \u2211' x, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum hX]\n\nlemma entropy_cond_eq_sum_finiteRange (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) [FiniteRange X]:\n H[X | Y \u2190 y ; \u03bc] = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum_finiteRange hX]\n\n/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\nlemma entropy_comp_of_injective\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 T) (hf : Function.Injective f) :\n H[f \u2218 X ; \u03bc] = H[X ; \u03bc] := by\n have hf_m : Measurable f := measurable_of_countable f\n rw [entropy_def, \u2190 Measure.map_map hf_m hX, measureEntropy_map_of_injective _ _ hf_m hf,\n entropy_def]\n\n\n/-- The entropy of any constant is zero. -/\n@[simp] lemma entropy_const [IsProbabilityMeasure \u03bc] (c : S) : H[fun _ \u21a6 c ; \u03bc] = 0 := by\n simp [entropy, Measure.map_const]\n\nopen Set\n\nopen Function\n\n/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\nlemma IsUniform.entropy_eq (H : Finset S) (X : \u03a9 \u2192 S) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n haveI : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX'.aemeasurable\n have (t : S) : negMulLog ((\u03bc.map X).real {t}) = ((\u03bc.map X).real {t}) * log (Nat.card H) := by\n by_cases ht : t \u2208 H\n \u00b7 simp only [negMulLog, neg_mul, neg_mul_eq_mul_neg, IsUniform.measureReal_preimage_of_mem'\n hX hX' ht, one_div, log_inv, neg_neg]\n \u00b7 simp only [negMulLog, map_measureReal_apply hX' (MeasurableSet.singleton t),\n IsUniform.measureReal_preimage_of_nmem hX ht, neg_zero, log_zero, mul_zero, zero_mul]\n rw [entropy_eq_sum_finset' (A := H) hX', Finset.sum_congr rfl (fun t _ \u21a6 this t), \u2190 Finset.sum_mul,\n Finset.sum_realMeasure_singleton]\n . convert one_mul _\n have := IsUniform.full_measure hX hX'\n simp at this\n rw [Measure.real, this]\n simp\n rw [Measure.map_apply hX' (by measurability)]\n exact hX.measure_preimage_compl\n\n\n/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\nlemma IsUniform.entropy_eq' {H : Set S} [Finite H] {X : \u03a9 \u2192 S} {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n convert IsUniform.entropy_eq H.toFinite.toFinset X ?_ hX' with x\n . simp; exact Iff.rfl\n . infer_instance\n convert hX\n simp\n\n/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\nlemma entropy_eq_log_card {X : \u03a9 \u2192 S} [Fintype S] (hX : Measurable X) (\u03bc : Measure \u03a9) [h\u03bc : NeZero \u03bc]\n [IsFiniteMeasure \u03bc] :\n entropy X \u03bc = log (Fintype.card S) \u2194 \u2200 s, \u03bc.map X {s} = \u03bc Set.univ / Fintype.card S := by\n rcases eq_zero_or_neZero (\u03bc.map X) with h | h\n . have := Measure.le_map_apply (@Measurable.aemeasurable \u03a9 S _ _ X \u03bc hX) Set.univ\n simp [h] at this; simp [this] at h\u03bc\n have : IsFiniteMeasure (\u03bc.map X) := by\n apply Measure.isFiniteMeasure_map\n rw [entropy_def, measureEntropy_eq_card_iff_measure_eq, Measure.map_apply hX MeasurableSet.univ]\n simp\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\nlemma prob_ge_exp_neg_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) [hX': FiniteRange X] :\n \u2203 s : S, \u03bc.map X {s} \u2265 (\u03bc Set.univ) * (rexp (- H[X ; \u03bc])).toNNReal := by\n let \u03bcS := \u03bc.map X\n let \u03bcs s := \u03bcS {s}\n rcases finiteSupport_of_finiteRange (X := X) with \u27e8A, hA\u27e9\n\n let S_nonzero := A.filter (fun s \u21a6 \u03bcs s \u2260 0)\n\n set norm := \u03bcS A with rw_norm\n have h_norm : norm = \u03bc Set.univ := by\n have := measure_add_measure_compl (\u03bc := \u03bcS) (s := A) (Finset.measurableSet _)\n rw [hA, add_zero] at this\n simp [norm, \u03bcS, this, Measure.map_apply hX MeasurableSet.univ]\n\n let pdf_nn s := norm\u207b\u00b9 * \u03bcs s\n let pdf s := (pdf_nn s).toReal\n let neg_log_pdf s := -log (pdf s)\n\n rcases Finset.eq_empty_or_nonempty S_nonzero with h_empty | h_nonempty\n . have h_norm_zero : \u03bc Set.univ = 0 := by\n have h : \u2200 s \u2208 A, \u03bcs s \u2260 0 \u2192 \u03bcs s \u2260 0 := fun _ _ h \u21a6 h\n rw [\u2190 h_norm, rw_norm, \u2190 Finset.sum_measure_singleton, \u2190 Finset.sum_filter_of_ne h,\n show Finset.filter _ _ = S_nonzero from rfl, h_empty, show Finset.sum \u2205 \u03bcs = 0 from rfl]\n use Classical.arbitrary (\u03b1 := S)\n rw [h_norm_zero, zero_mul]\n exact le_of_not_gt ENNReal.not_lt_zero\n\n rcases exists_or_forall_not (fun s \u21a6 \u03bc.map X {s} = \u221e) with h_infty | h_finite\n . obtain \u27e8s, h_s\u27e9 := h_infty\n use s; rw [h_s] ; exact le_top\n\n rcases eq_zero_or_neZero \u03bc with h_zero_measure | _\n . use Classical.arbitrary (\u03b1 := S)\n rw [h_zero_measure, show (0 : Measure \u03a9) _ = 0 from rfl, zero_mul]\n exact zero_le _\n\n have h_norm_pos : 0 < norm := by\n rw [h_norm, Measure.measure_univ_pos]\n exact NeZero.ne \u03bc\n have h_norm_finite : norm < \u221e := by\n rw [rw_norm, \u2190 Finset.sum_measure_singleton]\n exact ENNReal.sum_lt_top (fun s _ \u21a6 h_finite s)\n have h_invinvnorm_finite : norm\u207b\u00b9\u207b\u00b9 \u2260 \u221e := by\n rw [inv_inv]\n exact LT.lt.ne_top h_norm_finite\n have h_invnorm_ne_zero : norm\u207b\u00b9 \u2260 0 := ENNReal.inv_ne_top.mp h_invinvnorm_finite\n have h_invnorm_finite : norm\u207b\u00b9 \u2260 \u221e := by\n rw [\u2190 ENNReal.inv_ne_zero, inv_inv]\n exact ne_zero_of_lt h_norm_pos\n have h_pdf_finite : \u2200 s, pdf_nn s \u2260 \u221e := fun s \u21a6 ENNReal.mul_ne_top h_invnorm_finite (h_finite s)\n\n have h_norm_cancel : norm * norm\u207b\u00b9 = 1 :=\n ENNReal.mul_inv_cancel (ne_zero_of_lt h_norm_pos) (LT.lt.ne_top h_norm_finite)\n have h_pdf1 : (\u2211 s in A, pdf s) = 1 := by\n rw [\u2190 ENNReal.toReal_sum (fun s _ \u21a6 h_pdf_finite s), \u2190 Finset.mul_sum,\n Finset.sum_measure_singleton, mul_comm, h_norm_cancel, ENNReal.one_toReal]\n\n let \u27e8s_max, hs, h_min\u27e9 := Finset.exists_min_image S_nonzero neg_log_pdf h_nonempty\n have h_pdf_s_max_pos : 0 < pdf s_max := by\n rw [Finset.mem_filter] at hs\n have h_nonzero : pdf s_max \u2260 0 := ENNReal.toReal_ne_zero.mpr\n \u27e8mul_ne_zero h_invnorm_ne_zero hs.2, ENNReal.mul_ne_top h_invnorm_finite (h_finite s_max)\u27e9\n exact LE.le.lt_of_ne ENNReal.toReal_nonneg h_nonzero.symm\n\n use s_max\n rw [\u2190 h_norm, \u2190 one_mul (\u03bc.map X _), \u2190 h_norm_cancel, mul_assoc]\n apply mul_le_mul_of_nonneg_left _ (le_of_lt h_norm_pos)\n show ENNReal.ofReal (rexp (-H[X ; \u03bc])) \u2264 pdf_nn s_max\n rw [ENNReal.ofReal_le_iff_le_toReal (h_pdf_finite _),\n show (pdf_nn _).toReal = pdf _ from rfl, \u2190 Real.exp_log h_pdf_s_max_pos]\n apply exp_monotone\n rw [neg_le, \u2190 one_mul (-log _), \u2190 h_pdf1, Finset.sum_mul]\n let g_lhs s := pdf s * neg_log_pdf s_max\n let g_rhs s := -pdf s * log (pdf s)\n suffices \u2211 s in A, g_lhs s \u2264 \u2211 s in A, g_rhs s by\n convert this\n rw [entropy_eq_sum_finset hX hA]\n congr with s\n simp only [negMulLog, neg_mul, ENNReal.toReal_mul, neg_inj, g_rhs, pdf, pdf_nn]\n simp at h_norm\n rw [h_norm]\n simp\n have h_lhs : \u2200 s, \u03bcs s = 0 \u2192 g_lhs s = 0 := by {intros _ h; simp [g_lhs, pdf, pdf_nn, h]}\n have h_rhs : \u2200 s, \u03bcs s = 0 \u2192 g_rhs s = 0 := by {intros _ h; simp [g_rhs, pdf, pdf_nn, h]}\n rw [\u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_lhs s).mt),\n \u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_rhs s).mt)]\n apply Finset.sum_le_sum\n intros s h_s\n rw [show g_lhs s = _ * _ from rfl, show g_rhs s = _ * _ from rfl, neg_mul_comm]\n exact mul_le_mul_of_nonneg_left (h_min s h_s) ENNReal.toReal_nonneg\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\nlemma prob_ge_exp_neg_entropy' {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (X : \u03a9 \u2192 S) (hX : Measurable X) [FiniteRange X] :\n \u2203 s : S, rexp (- H[X ; \u03bc]) \u2264 \u03bc.real (X \u207b\u00b9' {s}) := by\n obtain \u27e8s, hs\u27e9 := prob_ge_exp_neg_entropy X \u03bc hX\n use s\n rwa [IsProbabilityMeasure.measure_univ, one_mul, ge_iff_le,\n (show ENNReal.ofNNReal _ = ENNReal.ofReal _ from rfl),\n ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _), \u2190 Measure.real,\n map_measureReal_apply hX (MeasurableSet.singleton s)] at hs\n\n/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\nlemma const_of_nonpos_entropy {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] {X : \u03a9 \u2192 S} (hX : Measurable X) [FiniteRange X] (hent: H[X; \u03bc] \u2264 0):\n \u2203 s : S, \u03bc.real (X \u207b\u00b9' {s}) = 1 := by\n rcases prob_ge_exp_neg_entropy' (\u03bc := \u03bc) X hX with \u27e8 s, hs \u27e9\n use s\n apply LE.le.antisymm\n . rw [\u2190 IsProbabilityMeasure.measureReal_univ (\u03bc := \u03bc)]\n exact measureReal_mono (subset_univ _) (by finiteness)\n refine le_trans ?_ hs\n simp [hent]\n\n\n/-- $H[X, Y] = H[Y, X]$. -/\nlemma entropy_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n change H[Prod.swap \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hY.prod_mk hX) Prod.swap Prod.swap_injective\n\n/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\nlemma entropy_assoc (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n change H[MeasurableEquiv.prodAssoc \u2218 \u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc ((hX.prod_mk hY).prod_mk hZ) _ $ Equiv.injective _\n\n/-- $H[X, f(X)] = H[X]$.-/\n@[simp] lemma entropy_prod_comp (hX : Measurable X) (\u03bc : Measure \u03a9) (f : S \u2192 T) :\n H[\u27e8X, f \u2218 X\u27e9; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) fun _ _ ab \u21a6 (Prod.ext_iff.1 ab).1\n\nend entropy\n\nsection condEntropy\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n\n/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\nnoncomputable\ndef condEntropy (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]]\n\nlemma condEntropy_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n condEntropy X Y \u03bc = (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]] := rfl\n\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \" ; \" \u03bc \"]\" => condEntropy X Y \u03bc\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \"]\" => condEntropy X Y volume\n\nlemma condEntropy_eq_zero\n (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (t : T) (ht : (\u03bc.map Y {t}).toReal = 0) : H[X | Y \u2190 t ; \u03bc] = 0 := by\n convert entropy_zero_measure X\n apply cond_eq_zero_of_meas_eq_zero\n rw [Measure.map_apply hY (MeasurableSet.singleton t)] at ht\n rw [\u2190 measureReal_eq_zero_iff]\n exact ht\n\n/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\nlemma condEntropy_eq_kernel_entropy\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] [FiniteRange Y] :\n H[X | Y ; \u03bc] = Hk[condDistrib X Y \u03bc, \u03bc.map Y] := by\n rw [condEntropy_def, kernel.entropy]\n apply integral_congr_finiteSupport\n intro t ht\n simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top (\u03bc.map Y), or_false] at ht\n rw [Measure.map_apply hY (measurableSet_singleton _)] at ht\n simp only [entropy_def]\n congr\n ext S hS\n rw [condDistrib_apply' hX hY _ _ ht hS, Measure.map_apply hX hS,\n cond_apply _ (hY (measurableSet_singleton _))]\n\n/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\nlemma map_prod_comap_swap (hX : Measurable X) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9))).comap Prod.swap = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext s hs\n rw [Measure.map_apply (hZ.prod_mk hX) hs, Measure.comap_apply _ Prod.swap_injective _ _ hs]\n \u00b7 rw [Measure.map_apply (hX.prod_mk hZ)]\n \u00b7 congr with \u03c9\n simp only [Set.image_swap_eq_preimage_swap, Set.mem_preimage, Prod.swap_prod_mk]\n \u00b7 exact MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' hs\n \u00b7 exact fun t ht \u21a6 MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' ht\n\nlemma condEntropy_two_eq_kernel_entropy (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] =\n Hk[kernel.condKernel (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc),\n Measure.map Z \u03bc \u2297\u2098 kernel.fst (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc)] := by\n rw [Measure.compProd_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc),\n map_compProd_condDistrib hY hZ,\n kernel.entropy_congr (condKernel_condDistrib_ae_eq hY hX hZ \u03bc),\n \u2190 kernel.entropy_congr (swap_condDistrib_ae_eq hY hX hZ \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, Y \u03c9)) = (\u03bc.map (fun \u03c9 \u21a6 (Y \u03c9, Z \u03c9))).comap Prod.swap := by\n rw [map_prod_comap_swap hY hZ]\n rw [this, condEntropy_eq_kernel_entropy hX (hY.prod_mk hZ), kernel.entropy_comap_swap]\n\n/-- Any random variable on a zero measure space has zero conditional entropy. -/\n@[simp]\nlemma condEntropy_zero_measure (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) : H[X | Y ; (0 : Measure \u03a9)] = 0 :=\n by simp [condEntropy]\n\n/-- Conditional entropy is non-negative. -/\nlemma condEntropy_nonneg (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) : 0 \u2264 H[X | Y ; \u03bc] :=\n integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\nlemma condEntropy_le_log_card [MeasurableSingletonClass S] [Fintype S]\n (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n H[X | Y ; \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_nonneg _ _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_le_log_card _ _)\n \u00b7 have : IsProbabilityMeasure (\u03bc.map Y) := isProbabilityMeasure_map hY.aemeasurable\n simp\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY: Measurable Y) [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 y in FiniteRange.toFinset Y, (\u03bc.map Y {y}).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum' (s := FiniteRange.toFinset Y) _ _]\n simp_rw [smul_eq_mul]\n exact full_measure_of_finiteRange hY\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum_fintype\n [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY : Measurable Y) [Fintype T] :\n H[X | Y ; \u03bc] = \u2211 y, (\u03bc (Y \u207b\u00b9' {y})).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum]\n simp_rw [smul_eq_mul, Measure.map_apply hY (measurableSet_singleton _)]\n\nlemma condEntropy_prod_eq_sum {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 T'} [MeasurableSpace T']\n [MeasurableSingletonClass T']\n (\u03bc : Measure \u03a9) (hY : Measurable Y) (hZ : Measurable Z)\n [IsFiniteMeasure \u03bc] [Fintype T] [Fintype T'] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc]\n = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc[|Z \u207b\u00b9' {z}]] := by\n simp_rw [condEntropy_eq_sum_fintype _ _ _ (hY.prod_mk hZ), condEntropy_eq_sum_fintype _ _ _ hY,\n Fintype.sum_prod_type_right, Finset.mul_sum, \u2190 mul_assoc, \u2190 ENNReal.toReal_mul]\n congr with y\n congr with x\n have A : (fun a \u21a6 (Y a, Z a)) \u207b\u00b9' {(x, y)} = Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x} := by\n ext p; simp [and_comm]\n congr 2\n \u00b7 rw [cond_apply _ (hZ (measurableSet_singleton y)), \u2190 mul_assoc, A]\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {y})) 0 with hy|hy\n \u00b7 have : \u03bc (Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x}) = 0 :=\n le_antisymm ((measure_mono (Set.inter_subset_left _ _)).trans hy.le) bot_le\n simp [this, hy]\n \u00b7 rw [ENNReal.mul_inv_cancel hy (by finiteness), one_mul]\n \u00b7 rw [A, cond_cond_eq_cond_inter _ (hZ (.singleton y)) (hY (.singleton x))]\n\n/-- $H[X|Y] = \\sum_y \\sum_x P[Y=y] P[X=x|Y=y] log \\frac{1}{P[X=x|Y=y]}$.-/\nlemma condEntropy_eq_sum_sum [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T} (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc]\n = \u2211 y in FiniteRange.toFinset Y, \u2211 x in FiniteRange.toFinset X, (\u03bc.map Y {y}).toReal * negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n rw [condEntropy_eq_sum _ _ _ hY]\n congr with y\n rw [entropy_cond_eq_sum_finiteRange hX, Finset.mul_sum]\n\n/-- Same as previous lemma, but with a sum over a product space rather than a double sum. -/\nlemma condEntropy_eq_sum_prod [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T}\n (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 p in (FiniteRange.toFinset X) \u00d7\u02e2 (FiniteRange.toFinset Y),\n (\u03bc.map Y {p.2}).toReal * negMulLog ((\u03bc[|Y \u207b\u00b9' {p.2}]).map X {p.1}).toReal := by\n rw [condEntropy_eq_sum_sum hX hY, Finset.sum_product_right]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$ are random variables, and $f : T \\times S \u2192 U$ is\n injective for each fixed $t \\in T$, then $H[f(Y, X)|Y] = H[X|Y]$.\n Thus for instance $H[X-Y|Y] = H[X|Y]$. -/\nlemma condEntropy_of_injective\n [MeasurableSingletonClass U] (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (hX : Measurable X)\n (hY : Measurable Y) (f : T \u2192 S \u2192 U) (hf : \u2200 t, Injective (f t)) [FiniteRange Y] :\n H[(fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)) | Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_eq_sum _ _ _ hY, condEntropy_eq_sum _ _ _ hY]\n have : \u2200 y, H[fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)|Y\u2190 y; \u03bc] = H[(f y \u2218 X) | Y \u2190 y ; \u03bc] := by\n intro y\n refine entropy_congr ?_\n have : \u2200\u1d50 \u03c9 \u2202\u03bc[|Y \u2190 y], Y \u03c9 = y := by\n rw [ae_iff, cond_apply _ (hY (measurableSet_singleton _))]\n have : {a | \u00acY a = y} = (Y \u207b\u00b9' {y})\u1d9c := by ext; simp\n rw [this, Set.inter_compl_self, measure_empty, mul_zero]\n filter_upwards [this] with \u03c9 h\u03c9\n rw [h\u03c9]\n simp\n simp_rw [this]\n congr with y\n rw [entropy_comp_of_injective _ hX (f y) (hf y)]\n\n/-- A weaker version of the above lemma in which f is independent of Y. -/\nlemma condEntropy_comp_of_injective [MeasurableSingletonClass S] [MeasurableSingletonClass U]\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 U) (hf : Injective f) :\n H[f \u2218 X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n integral_congr_ae (ae_of_all _ (fun _ \u21a6 entropy_comp_of_injective _ hX f hf))\n\n/-- $H[X, Y| Z] = H[Y, X| Z]$. -/\nlemma condEntropy_comm {Z : \u03a9 \u2192 U}\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[\u27e8Y, X\u27e9 | Z; \u03bc] := by\n change H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Prod.swap \u2218 \u27e8X, Y\u27e9 | Z; \u03bc]\n exact (condEntropy_comp_of_injective \u03bc (hX.prod_mk hY) Prod.swap Prod.swap_injective).symm\n\nend condEntropy\n\nsection pair\n\n/-- One form of the chain rule : $H[X, Y] = H[X] + H[Y|X]. -/\nlemma chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y]:\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y | X ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n rw [entropy_eq_kernel_entropy, kernel.chain_rule]\n simp_rw [\u2190 kernel.map_const _ (hX.prod_mk hY), kernel.fst_map_prod _ hX hY, kernel.map_const _ hX,\n kernel.map_const _ (hX.prod_mk hY)]\n . congr 1\n \u00b7 rw [kernel.entropy, integral_dirac]\n rfl\n \u00b7 simp_rw [condEntropy_eq_kernel_entropy hY hX]\n have : Measure.dirac () \u2297\u2098 kernel.const Unit (\u03bc.map X) = \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) := by\n ext s _\n rw [Measure.dirac_unit_compProd_const, Measure.map_map measurable_prod_mk_left hX]\n congr\n rw [this, kernel.entropy_congr (condDistrib_const_unit hX hY \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) = (\u03bc.map X).map (Prod.mk ()) := by\n ext s _\n rw [Measure.map_map measurable_prod_mk_left hX]\n rfl\n rw [this, kernel.entropy_prodMkLeft_unit]\n \u00b7 apply kernel.FiniteKernelSupport.aefiniteKernelSupport\n exact kernel.finiteKernelSupport_of_const _\n\n/-- Another form of the chain rule : $H[X, Y] = H[Y] + H[X|Y]$. -/\nlemma chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[Y ; \u03bc] + H[X | Y ; \u03bc] := by\n rw [entropy_comm hX hY, chain_rule' \u03bc hY hX]\n\n/-- Another form of the chain rule : $H[X|Y] = H[X, Y] - H[Y]$. -/\nlemma chain_rule'' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] - H[Y ; \u03bc] := by\n rw [chain_rule \u03bc hX hY, add_sub_cancel_left]\n\n/-- Two pairs of variables that have the same joint distribution, have the same\nconditional entropy. -/\nlemma IdentDistrib.condEntropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y')\n (h : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') [FiniteRange X] [FiniteRange Y] [FiniteRange X']\n [FiniteRange Y'] :\n H[X | Y ; \u03bc] = H[X' | Y' ; \u03bc'] := by\n have : IdentDistrib Y Y' \u03bc \u03bc' := h.comp measurable_snd\n rw [chain_rule'' _ hX hY, chain_rule'' _ hX' hY', h.entropy_eq, this.entropy_eq]\n\n/-- If $X : \\Omega \\to S$ and $Y : \\Omega \\to T$ are random variables, and $f : T \\to U$ is an\ninjection then $H[X|f(Y)] = H[X|Y]$.\n -/\nlemma condEntropy_of_injective' [MeasurableSingletonClass S] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) (f : T \u2192 U) (hf : Injective f)\n (hfY : Measurable (f \u2218 Y)) [FiniteRange X] [FiniteRange Y] :\n H[X | f \u2218 Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [chain_rule'' \u03bc hX hY, chain_rule'' \u03bc hX hfY, chain_rule' \u03bc hX hY, chain_rule' \u03bc hX hfY]\n congr 1\n . congr 1\n exact condEntropy_comp_of_injective \u03bc hY f hf\n exact entropy_comp_of_injective \u03bc hY f hf\n\n/-- $H[X|f(X)] = H[X] - H[f(X)]$. -/\nlemma condEntropy_comp_self [IsProbabilityMeasure \u03bc] (hX : Measurable X) {f : S \u2192 U}\n (hf : Measurable f) [FiniteRange X] : H[X | f \u2218 X ; \u03bc] = H[X ; \u03bc] - H[f \u2218 X ; \u03bc] := by\n rw [chain_rule'' \u03bc hX (hf.comp hX), entropy_prod_comp hX _ f]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$,$Z : \\Omega \\to U$ are random variables, then\n$$H[X, Y | Z] = H[X | Z] + H[Y|X, Z]$$. -/\nlemma cond_chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[X | Z ; \u03bc] + H[Y | \u27e8X, Z\u27e9 ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condEntropy_eq_kernel_entropy (hX.prod_mk hY) hZ, kernel.chain_rule]\n . congr 1\n . rw [condEntropy_eq_kernel_entropy hX hZ]\n refine kernel.entropy_congr ?_\n exact condDistrib_fst_ae_eq hX hY hZ \u03bc\n \u00b7 rw [condEntropy_two_eq_kernel_entropy hY hX hZ]\n exact kernel.aefiniteKernelSupport_condDistrib _ _ \u03bc (by measurability) (by measurability)\n\n/-- $$ H[X, Y | Z] = H[Y | Z] + H[X|Y, Z].$$ -/\nlemma cond_chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Y | Z ; \u03bc] + H[X | \u27e8Y, Z\u27e9 ; \u03bc] := by\n rw [condEntropy_comm hX hY, cond_chain_rule' _ hY hX hZ]\n\n/-- Data-processing inequality for the entropy :\n$$ H[f(X)] \\leq H[X].$$\nTo upgrade this to equality, see `entropy_of_comp_eq_of_comp` or `entropy_comp_of_injective`. -/\nlemma entropy_comp_le (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (f : S \u2192 U) [FiniteRange X]:\n H[f \u2218 X ; \u03bc] \u2264 H[X ; \u03bc] := by\n have hfX : Measurable (f \u2218 X) := (measurable_of_countable _).comp hX\n have : H[X ; \u03bc] = H[\u27e8X, f \u2218 X\u27e9 ; \u03bc] := by\n refine (entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) ?_).symm\n intro x y hxy\n simp only [Prod.mk.injEq] at hxy\n exact hxy.1\n rw [this, chain_rule _ hX hfX]\n simp only [le_add_iff_nonneg_right]\n exact condEntropy_nonneg X (f \u2218 X) \u03bc\n\n/-- A Schroder-Bernstein type theorem for entropy : if two random variables are functions of each\n other, then they have the same entropy. Can be used as a substitute for\n `entropy_comp_of_injective` if one doesn't want to establish the injectivity. -/\nlemma entropy_of_comp_eq_of_comp\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n (f : S \u2192 T) (g : T \u2192 S) (h1 : Y = f \u2218 X) (h2 : X = g \u2218 Y) [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] = H[Y ; \u03bc] := by\n have h3 : H[X ; \u03bc] \u2264 H[Y ; \u03bc] := by\n rw [h2]; exact entropy_comp_le \u03bc hY _\n have h4 : H[Y ; \u03bc] \u2264 H[X ; \u03bc] := by\n rw [h1]; exact entropy_comp_le \u03bc hX _\n linarith\n\nend pair\n\nsection mutualInfo\n\n/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\nnoncomputable\ndef mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc]\n\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \" ; \" \u03bc \"]\" => mutualInfo X Y \u03bc\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \"]\" => mutualInfo X Y volume\n\nlemma mutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc] := rfl\n\n/-- $I[X : Y] = I[Y : X]$. -/\nlemma mutualInfo_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = I[Y : X ; \u03bc] := by simp_rw [mutualInfo, add_comm, entropy_comm hX hY]\n\nlemma entropy_add_entropy_sub_mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] + H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := sub_sub_self _ _\n\n/-- $I[X : Y] = H[X] - H[X|Y]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[X ; \u03bc] - H[X | Y ; \u03bc] := by\n rw [mutualInfo_def, chain_rule \u03bc hX hY]\n abel\n\n/-- $I[X : Y] = H[Y] - H[Y | X]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[Y ; \u03bc] - H[Y | X ; \u03bc] := by\n rw [mutualInfo_comm hX hY, mutualInfo_eq_entropy_sub_condEntropy hY hX]\n\n/-- $H[X] - I[X : Y] = H[X | Y]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] - I[X : Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY, sub_sub_self]\n\n/-- $H[Y] - I[X : Y] = H[Y | X]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[Y | X ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy' hX hY, sub_sub_self]\n\n/-- Mutual information is non-negative. -/\nlemma mutualInfo_nonneg (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [FiniteRange X] [FiniteRange Y] :\n 0 \u2264 I[X : Y ; \u03bc] := by\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n exact measureMutualInfo_nonneg\n\n/-- Substituting variables for ones with the same distributions doesn't change the mutual information. -/\nlemma IdentDistrib.mutualInfo_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} (hXY : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') :\n I[X : Y ; \u03bc] = I[X' : Y' ; \u03bc'] := by\n have hX : IdentDistrib X X' \u03bc \u03bc' := hXY.comp measurable_fst\n have hY : IdentDistrib Y Y' \u03bc \u03bc' := hXY.comp measurable_snd\n simp_rw [mutualInfo_def,hX.entropy_eq,hY.entropy_eq,hXY.entropy_eq]\n\n/-- Subadditivity of entropy. -/\nlemma entropy_pair_le_add (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] \u2264 H[X ; \u03bc] + H[Y ; \u03bc] :=\n sub_nonneg.1 $ mutualInfo_nonneg hX hY _\n\n/-- $I[X : Y] = 0$ iff $X, Y$ are independent. -/\nlemma mutualInfo_eq_zero (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n I[X : Y ; \u03bc] = 0 \u2194 IndepFun X Y \u03bc := by\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n convert measureMutualInfo_eq_zero_iff (\u03bc := \u03bc.map (\u27e8X, Y\u27e9))\n rw [indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable,\n Measure.ext_iff_measureReal_singleton_finiteSupport]\n congr! with p\n convert measureReal_prod_prod (\u03bc := \u03bc.map X) (\u03bd := \u03bc.map Y) {p.1} {p.2}\n \u00b7 simp\n \u00b7 exact Measure.map_map measurable_fst (hX.prod_mk hY)\n \u00b7 exact Measure.map_map measurable_snd (hX.prod_mk hY)\n\nprotected alias \u27e8_, IndepFun.mutualInfo_eq_zero\u27e9 := mutualInfo_eq_zero\n\n/-- The mutual information with a constant is always zero. -/\nlemma mutualInfo_const (hX : Measurable X) (c : T) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X] :\n I[X : fun _ \u21a6 c ; \u03bc] = 0 := (indepFun_const c).mutualInfo_eq_zero hX measurable_const\n\nlemma IndepFun.condEntropy_eq_entropy {\u03bc : Measure \u03a9} (h : IndepFun X Y \u03bc)\n (hX : Measurable X) (hY : Measurable Y) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[X ; \u03bc] := by\n have := h.mutualInfo_eq_zero hX hY\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY] at this\n linarith\n\n/-- $H[X, Y] = H[X] + H[Y]$ if and only if $X, Y$ are independent. -/\nlemma entropy_pair_eq_add (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] \u2194 IndepFun X Y \u03bc := by\n rw [eq_comm, \u2190 sub_eq_zero, \u2190 mutualInfo_eq_zero hX hY]; rfl\n\n/-- If $X, Y$ are independent, then $H[X, Y] = H[X] + H[Y]$. -/\nprotected alias \u27e8_, IndepFun.entropy_pair_eq_add\u27e9 := entropy_pair_eq_add\n\n/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\nnoncomputable\ndef condMutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9 := by volume_tac) :\n \u211d := (\u03bc.map Z)[fun z \u21a6 H[X | Z \u2190 z ; \u03bc] + H[Y | Z \u2190 z ; \u03bc] - H[\u27e8X, Y\u27e9 | Z \u2190 z ; \u03bc]]\n\nlemma condMutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9) :\n condMutualInfo X Y Z \u03bc = (\u03bc.map Z)[fun z \u21a6\n H[X | Z \u2190 z ; \u03bc] + H[Y | Z \u2190 z ; \u03bc] - H[\u27e8X, Y\u27e9 | Z \u2190 z ; \u03bc]] := rfl\n\n@[inherit_doc condMutualInfo]\nnotation3:max \"I[\" X \" : \" Y \"|\" Z \";\" \u03bc \"]\" => condMutualInfo X Y Z \u03bc\n@[inherit_doc condMutualInfo]\nnotation3:max \"I[\" X \" : \" Y \"|\" Z \"]\" => condMutualInfo X Y Z volume\n\n", "theoremStatement": "/-- The conditional mutual information agrees with the information of the conditional kernel.\n-/\nlemma condMutualInfo_eq_kernel_mutualInfo\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = Ik[condDistrib (\u27e8X, Y\u27e9) Z \u03bc, \u03bc.map Z] ", "theoremName": "ProbabilityTheory.condMutualInfo_eq_kernel_mutualInfo", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "814b2e74d08a5c193829e5f831b3b8c40bfb8dbf", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 768, "tokenPositionInFile": 37470, "theoremPositionInFile": 82}, "dependencyMetadata": {"inFilePremises": true, 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"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.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.Lemma", "Mathlib.Tactic.TypeStar", "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.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.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.Init.Data.Int.Basic", "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.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "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.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.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.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "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.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.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rcases finiteSupport_of_finiteRange (\u03bc := \u03bc) (X := Z) with \u27e8A, hA\u27e9\n simp_rw [condMutualInfo_def, entropy_def, kernel.mutualInfo, kernel.entropy,\n integral_eq_sum' _ hA, smul_eq_mul, mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib]\n congr with x\n \u00b7 have h := condDistrib_fst_ae_eq hX hY hZ \u03bc\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n specialize h x\n by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [h hx, condDistrib_apply hX hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n \u00b7 have h := condDistrib_snd_ae_eq hX hY hZ \u03bc\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n specialize h x\n by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [h hx, condDistrib_apply hY hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n \u00b7 by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [condDistrib_apply (hX.prod_mk hY) hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx", "proofType": "tactic", "proofLengthLines": 22, "proofLengthTokens": 991}} +{"srcContext": "import PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The fibring identity\n\nThe proof of the fibring identity, which is a key component of the proof of PFR.\n\n## Main statement\n\n* `sum_of_rdist_eq`: If $Y_1,Y_2,Y_3,Y_4$ are independent, then $d[Y_1; Y_2] + d[Y_3; Y_4]$ is equal to the sum of\n$$d[Y_1+Y_3; Y_2+Y_4] + d[Y_1|Y_1+Y_3; Y_2|Y_2+Y_4] $$\nand\n$$ I[Y_1+Y_2 : Y_2 + Y_4 | Y_1+Y_2+Y_3+Y_4].$$\n\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralFibring\n\n-- $\\pi : H \\to H'$ is a homomorphism between additive groups.\nvariable {H : Type*} [AddCommGroup H] [Countable H] [hH : MeasurableSpace H]\n [MeasurableSingletonClass H]\n {H' : Type*} [AddCommGroup H'] [Countable H'] [hH' : MeasurableSpace H']\n [MeasurableSingletonClass H']\n (\u03c0 : H \u2192+ H')\nvariable {\u03a9 \u03a9' : Type*} [m\u03a9 : MeasurableSpace \u03a9] [m\u03a9' : MeasurableSpace \u03a9']\n {\u03bc : Measure \u03a9} {\u03bc' : Measure \u03a9'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $Z_1, Z_2$ are independent, then $d[Z_1; Z_2]$ is equal to\n$$ d[\\pi(Z_1);\\pi(Z_2)] + d[Z_1|\\pi(Z_1); Z_2 |\\pi(Z_2)]$$\nplus\n$$I( Z_1 - Z_2 : (\\pi(Z_1), \\pi(Z_2)) | \\pi(Z_1 - Z_2) ).$$\n-/\nlemma rdist_of_indep_eq_sum_fibre {Z_1 Z_2: \u03a9 \u2192 H} (h : IndepFun Z_1 Z_2 \u03bc)\n (h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2]:\n d[Z_1; \u03bc # Z_2; \u03bc] = d[\u03c0 \u2218 Z_1; \u03bc # \u03c0 \u2218 Z_2; \u03bc] + d[Z_1|\u03c0\u2218Z_1; \u03bc # Z_2|\u03c0\u2218Z_2; \u03bc] + I[Z_1-Z_2 : \u27e8\u03c0\u2218Z_1, \u03c0\u2218Z_2\u27e9 | \u03c0\u2218(Z_1 - Z_2); \u03bc] := by\n have h\u03c0 : Measurable \u03c0 := measurable_of_countable _\n have step1 : d[Z_1; \u03bc # Z_2; \u03bc] = d[\u03c0 \u2218 Z_1; \u03bc # \u03c0 \u2218 Z_2; \u03bc] +\n H[(Z_1 - Z_2)| \u03c0 \u2218 (Z_1 - Z_2); \u03bc] - H[Z_1 | \u03c0 \u2218 Z_1; \u03bc] / 2 - H[Z_2 | \u03c0 \u2218 Z_2; \u03bc] / 2 := by\n have hsub : H[(Z_1 - Z_2)| \u03c0 \u2218 (Z_1 - Z_2); \u03bc] = H[(Z_1 - Z_2); \u03bc] - H[\u03c0 \u2218 (Z_1 - Z_2); \u03bc] := condEntropy_comp_self (by measurability) h\u03c0\n\n rw [h.rdist_eq h1 h2, (h.comp h\u03c0 h\u03c0).rdist_eq (h\u03c0.comp h1) (h\u03c0.comp h2),\n condEntropy_comp_self h1 h\u03c0, condEntropy_comp_self h2 h\u03c0, hsub, map_comp_sub \u03c0]\n ring_nf\n have m0 : Measurable (fun x \u21a6 (x, \u03c0 x)) := measurable_of_countable _\n have h' : IndepFun (\u27e8Z_1, \u03c0 \u2218 Z_1\u27e9) (\u27e8Z_2, \u03c0 \u2218 Z_2\u27e9) \u03bc := h.comp m0 m0\n have m1 : Measurable (Z_1 - Z_2) := h1.sub h2\n have m2 : Measurable (\u27e8\u2191\u03c0 \u2218 Z_1, \u2191\u03c0 \u2218 Z_2\u27e9) := (h\u03c0.comp h1).prod_mk (h\u03c0.comp h2)\n have m3 : Measurable (\u2191\u03c0 \u2218 (Z_1 - Z_2)) := h\u03c0.comp m1\n have entroplem : H[Z_1 - Z_2|\u27e8\u27e8\u2191\u03c0 \u2218 Z_1, \u2191\u03c0 \u2218 Z_2\u27e9, \u2191\u03c0 \u2218 (Z_1 - Z_2)\u27e9; \u03bc]\n = H[Z_1 - Z_2|\u27e8\u2191\u03c0 \u2218 Z_1, \u2191\u03c0 \u2218 Z_2\u27e9; \u03bc] := by\n rw [map_comp_sub \u03c0]\n let f : H' \u00d7 H' \u2192 (H' \u00d7 H') \u00d7 H' := fun (x,y) \u21a6 ((x,y), x - y)\n have hf : Function.Injective f := fun _ _ h \u21a6 (Prod.ext_iff.1 h).1\n have mf : Measurable f := measurable_id.prod_mk measurable_sub\n refine condEntropy_of_injective' \u03bc m1 m2 f hf (mf.comp m2)\n rw [step1, condMutualInfo_eq' m1 m2 m3, entroplem,\n condRuzsaDist_of_indep h1 (h\u03c0.comp h1) h2 (h\u03c0.comp h2) \u03bc h']\n ring_nf\n\n", "theoremStatement": "lemma rdist_le_sum_fibre {Z_1: \u03a9 \u2192 H} {Z_2: \u03a9' \u2192 H}\n (h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2] :\n d[\u03c0 \u2218 Z_1; \u03bc # \u03c0 \u2218 Z_2; \u03bc'] + d[Z_1|\u03c0\u2218Z_1; \u03bc # Z_2|\u03c0\u2218Z_2; \u03bc'] \u2264 d[Z_1; \u03bc # Z_2; \u03bc']", "theoremName": "rdist_le_sum_fibre", "fileCreated": {"commit": 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"Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8\u03bd, W_1, W_2, h\u03bd, m1, m2, hi, hi1, hi2, _, _\u27e9 := ProbabilityTheory.independent_copies_finiteRange h1 h2 \u03bc \u03bc'\n have h\u03c0 : Measurable \u03c0 := measurable_of_countable _\n have h\u03c6 : Measurable (fun x \u21a6 (x, \u03c0 x)) := measurable_of_countable _\n have h\u03c01 : IdentDistrib (\u27e8Z_1, \u03c0 \u2218 Z_1\u27e9) (\u27e8W_1, \u03c0 \u2218 W_1\u27e9) \u03bc \u03bd := hi1.symm.comp h\u03c6\n have h\u03c02 : IdentDistrib (\u27e8Z_2, \u03c0 \u2218 Z_2\u27e9) (\u27e8W_2, \u03c0 \u2218 W_2\u27e9) \u03bc' \u03bd := hi2.symm.comp h\u03c6\n rw [\u2190 hi1.rdist_eq hi2, \u2190 (hi1.comp h\u03c0).rdist_eq (hi2.comp h\u03c0),\n rdist_of_indep_eq_sum_fibre \u03c0 hi m1 m2,\n condRuzsaDist_of_copy h1 (h\u03c0.comp h1) h2 (h\u03c0.comp h2) m1 (h\u03c0.comp m1) m2 (h\u03c0.comp m2) h\u03c01 h\u03c02]\n exact le_add_of_nonneg_right (condMutualInfo_nonneg (by measurability) (Measurable.prod_mk (h\u03c0.comp m1) (h\u03c0.comp m2)) _ _)", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 751}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\nopen ElementaryAddCommGroup\n\n/-- In fact $k$ is at most\n $$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\nlemma construct_good_improved' :\n k \u2264 \u03b4 + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 # T\u2081 | T\u2082] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2082 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2082 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2083 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2083 | T\u2082] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2081 | T\u2082] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2081 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2083 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2083 | T\u2082] - d[p.X\u2080\u2082 # X\u2082])) := by\n have I1 : I[T\u2082 : T\u2081] = I[T\u2081 : T\u2082] := mutualInfo_comm hT\u2082 hT\u2081 _\n have I2 : I[T\u2083 : T\u2081] = I[T\u2081 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2081 _\n have I3 : I[T\u2083 : T\u2082] = I[T\u2082 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2082 _\n have Z123 := construct_good_prelim' h_min hT hT\u2081 hT\u2082 hT\u2083\n have h132 : T\u2081 + T\u2083 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z132 := construct_good_prelim' h_min h132 hT\u2081 hT\u2083 hT\u2082\n have h213 : T\u2082 + T\u2081 + T\u2083 = 0 := by rw [\u2190 hT]; abel\n have Z213 := construct_good_prelim' h_min h213 hT\u2082 hT\u2081 hT\u2083\n have h231 : T\u2082 + T\u2083 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z231 := construct_good_prelim' h_min h231 hT\u2082 hT\u2083 hT\u2081\n have h312 : T\u2083 + T\u2081 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z312 := construct_good_prelim' h_min h312 hT\u2083 hT\u2081 hT\u2082\n have h321 : T\u2083 + T\u2082 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z321 := construct_good_prelim' h_min h321 hT\u2083 hT\u2082 hT\u2081\n simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321\n linarith\n\n/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\nlemma construct_good_improved'' {\u03a9' : Type*} [MeasurableSpace \u03a9'] (\u03bc : Measure \u03a9')\n [IsProbabilityMeasure \u03bc] {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G}\n (hT : T\u2081 + T\u2082 + T\u2083 = 0) (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083) :\n k \u2264 I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc] + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]))\n := by\n let M : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n exact construct_good_improved' h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend aux\n\nopen scoped BigOperators\n\n/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\nlemma averaged_construct_good : k \u2264 (I[U : V | S] + I[V : W | S] + I[W : U | S])\n + (p.\u03b7 / 6) * (((d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]))\n + ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])))\n := by\n have hS : Measurable S := ((hX\u2081.add' hX\u2082).add' hX\u2081').add' hX\u2082'\n have hU : Measurable U := hX\u2081.add' hX\u2082\n have hV : Measurable V := hX\u2081'.add hX\u2082\n have hW : Measurable W := hX\u2081'.add hX\u2081\n have hUVW : U + V + W = 0 := sum_uvw_eq_zero X\u2081 X\u2082 X\u2081'\n have hz (a : \u211d) : a = \u2211 z, (\u2119 (S \u207b\u00b9' {z})).toReal * a := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u2119 hS, one_mul]\n rw [hz k, hz (d[p.X\u2080\u2081 # X\u2081]), hz (d[p.X\u2080\u2082 # X\u2082])]\n simp only [condMutualInfo_eq_sum' hS, \u2190 Finset.sum_add_distrib, \u2190 mul_add,\n condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, \u2190 Finset.sum_sub_distrib, \u2190 mul_sub, Finset.mul_sum,\n \u2190 mul_assoc (p.\u03b7/6), mul_comm (p.\u03b7/6), mul_assoc _ _ (p.\u03b7/6)]\n rw [Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_le_sum (fun i _hi \u21a6 ?_)\n rcases eq_or_ne (\u2119 (S \u207b\u00b9' {i})) 0 with h'i|h'i\n \u00b7 simp [h'i]\n rw [mul_assoc, \u2190 mul_add]\n gcongr\n have : IsProbabilityMeasure (\u2119[|S \u207b\u00b9' {i}]) := cond_isProbabilityMeasure \u2119 h'i\n linarith [construct_good_improved'' h_min (\u2119[|S \u207b\u00b9' {i}]) hUVW hU hV hW]\n\nvariable (p)\n\nlemma dist_diff_bound_1 :\n (d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n \u2264 (16 * k + 6 * d[X\u2081 # X\u2081] + 2 * d[X\u2082 # X\u2082]) / 4 + (H[X\u2081 + X\u2081'] - H[X\u2082 + X\u2082']) / 4\n + (H[X\u2082 | X\u2082 + X\u2082'] - H[X\u2081 | X\u2081 + X\u2081']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2082' X\u2081' hX\u2081 hX\u2082 hX\u2082' hX\u2081' h_indep.reindex_four_abcd\n have I2 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep.reindex_four_abdc\n have I3 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2082 X\u2081' hX\u2081 hX\u2082' hX\u2082 hX\u2081' h_indep.reindex_four_acbd\n have I4 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2081' X\u2082 hX\u2081 hX\u2082' hX\u2081' hX\u2082 h_indep.reindex_four_acdb\n have I5 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082 X\u2082' hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_adbc\n have I6 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082' X\u2082 hX\u2081 hX\u2081' hX\u2082' hX\u2082 h_indep.reindex_four_adcb\n have C1 : U + X\u2082' + X\u2081' = S := by abel\n have C2 : W + X\u2082 + X\u2082' = S := by abel\n have C3 : X\u2081 + X\u2082' + X\u2082 + X\u2081' = S := by abel\n have C4 : X\u2081 + X\u2082' + X\u2081' + X\u2082 = S := by abel\n have C5 : W + X\u2082' + X\u2082 = S := by abel\n have C7 : X\u2082 + X\u2081' = V := by abel\n have C8 : X\u2081 + X\u2081' = W := by abel\n have C9 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082.symm\n have C10 : d[X\u2082 # X\u2081'] = d[X\u2081' # X\u2082] := rdist_symm\n have C11 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081.symm\n have C12 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq h\u2082.symm\n have C13 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] := (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082.symm\n have C14 : d[X\u2081' # X\u2082] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C15 : H[X\u2081' + X\u2082'] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C16 : H[X\u2082'] = H[X\u2082] := h\u2082.symm.entropy_eq\n have C17 : H[X\u2081'] = H[X\u2081] := h\u2081.symm.entropy_eq\n have C18 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C19 : H[X\u2082' + X\u2081'] = H[U] := by rw [add_comm]; exact C15\n have C20 : d[X\u2082' # X\u2082] = d[X\u2082 # X\u2082] := h\u2082.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C21 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C22 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C23 : X\u2082' + X\u2082 = X\u2082 + X\u2082' := by abel\n have C24 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C25 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C27 : H[X\u2082' | X\u2082 + X\u2082'] = H[X\u2082 | X\u2082 + X\u2082'] := by\n conv_lhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2082'.add' hX\u2082) hX\u2082 (hX\u2082.add' hX\u2082')\n have I : IdentDistrib (\u27e8X\u2082', X\u2082\u27e9) (\u27e8X\u2082, X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2082.symm h\u2082\n (h_indep.indepFun (show 2 \u2260 1 by decide)) (h_indep.indepFun (show 1 \u2260 2 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C28 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C29 : H[X\u2081' | V] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082) hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C30 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n simp only [C1, C2, C3, C4, C5, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30] at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma dist_diff_bound_2 :\n ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]))\n \u2264 (16 * k + 6 * d[X\u2082 # X\u2082] + 2 * d[X\u2081 # X\u2081]) / 4 + (H[X\u2082 + X\u2082'] - H[X\u2081 + X\u2081']) / 4\n + (H[X\u2081 | X\u2081 + X\u2081'] - H[X\u2082 | X\u2082 + X\u2082']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2082' X\u2081' hX\u2082 hX\u2081 hX\u2082' hX\u2081' h_indep.reindex_four_bacd\n have I2 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2081' X\u2082' hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_badc\n have I3 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081 X\u2081' hX\u2082 hX\u2082' hX\u2081 hX\u2081' h_indep.reindex_four_bcad\n have I4 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081' X\u2081 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bcda\n have I5 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2081 X\u2082' hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bdac\n have I6 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2082' X\u2081 hX\u2082 hX\u2081' hX\u2082' hX\u2081 h_indep.reindex_four_bdca\n have C1 : X\u2082 + X\u2081 = X\u2081 + X\u2082 := by abel\n have C2 : X\u2081 + X\u2081' = W := by abel\n have C3 : U + X\u2082' + X\u2081' = S := by abel\n have C4 : X\u2082 + X\u2081' = V := by abel\n have C5 : X\u2082 + X\u2082' + X\u2081 + X\u2081' = S := by abel\n have C6 : X\u2082 + X\u2082' + X\u2081' + X\u2081 = S := by abel\n have C7 : V + X\u2081 + X\u2082' = S := by abel\n have C8 : V + X\u2082' + X\u2081 = S := by abel\n have C9 : d[X\u2082 # X\u2081] = d[X\u2081 # X\u2082] := rdist_symm\n have C10 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n have C11 : d[X\u2082 # X\u2081'] = d[X\u2081 # X\u2082] := by\n rw [rdist_symm]\n exact ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n have C12 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C13 : d[X\u2082' # X\u2081] = d[X\u2081 # X\u2082'] := rdist_symm\n have C14 : d[X\u2081' # X\u2081] = d[X\u2081 # X\u2081'] := rdist_symm\n have C15 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n have C16 : H[X\u2081' + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C17 : H[X\u2082' + X\u2081'] = H[X\u2081 + X\u2082] := by rw [add_comm]; exact C16\n have C18 : H[X\u2081'] = H[X\u2081] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2081.symm\n have C19 : H[X\u2082'] = H[X\u2082] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2082.symm\n have C20 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C21 : H[X\u2081' | W] = H[X\u2081 | W] := by\n conv_rhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2081) hX\u2081 (hX\u2081.add' hX\u2081')\n have I : IdentDistrib (\u27e8X\u2081', X\u2081\u27e9) (\u27e8X\u2081, X\u2081'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2081\n (h_indep.indepFun (show 3 \u2260 0 by decide)) (h_indep.indepFun (show 0 \u2260 3 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C22 : H[X\u2082' | X\u2082' + X\u2081] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C23 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C24 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C25 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C27 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n have C28 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C29 : H[X\u2082' + X\u2081] = H[X\u2081 + X\u2082] := by\n rw [add_comm]\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C30 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2081.symm\n have C31 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable) h\u2082.symm\n simp only [C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C25, C26, C27, C28, C29, C30, C31]\n at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma averaged_final : k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) := by\n apply (averaged_construct_good hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_min).trans\n have : 0 \u2264 p.\u03b7 := p.h\u03b7.le\n have := sum_condMutual_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep.reindex_four_abdc h_min\n gcongr ?_ + (p.\u03b7 / 6) * ?_\n linarith [dist_diff_bound_1 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep,\n dist_diff_bound_2 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep]\n\n", "theoremStatement": "/-- Suppose $0 < \\eta < 1/8$. Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$. The proof\nof this lemma uses copies `X\u2081', X\u2082'` already in the context. For a version that does not assume\nthese are given and constructs them instead, use `tau_strictly_decreases'`.\n-/\ntheorem tau_strictly_decreases_aux' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 ", "theoremName": "tau_strictly_decreases_aux'", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "d7b6033604422555584766a86b127ec46025a905", "date": "2023-12-15"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 669, "tokenPositionInFile": 36571, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 13, "numPremises": 312, "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", 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"Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "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.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.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.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.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.Paths", "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", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "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", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "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", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "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", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "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", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "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.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "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", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "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.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.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "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.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.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.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "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.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "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.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "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.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : 0 < p.\u03b7 := p.h\u03b7\n have : k \u2264 8 * p.\u03b7 * k := by calc\n k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) :=\n averaged_final p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n _ \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) +\n p.\u03b7 / 6 * (8 * k + 2 * (2 * (k + (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7)))) := by\n gcongr\n exact second_estimate_aux p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082\n h_indep.reindex_four_abdc h_min\n _ = 8 * p.\u03b7 * k - (1 - 5 * p.\u03b7 - 4/6 * p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7) := by ring\n _ \u2264 8 * p.\u03b7 * k := by\n simp only [tsub_le_iff_right, le_add_iff_nonneg_right]\n apply div_nonneg _ (by linarith)\n apply mul_nonneg (by linarith) _\n linarith [first_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min]\n apply le_antisymm _ (rdist_nonneg hX\u2081 hX\u2082)\n nlinarith", "proofType": "tactic", "proofLengthLines": 18, "proofLengthTokens": 938}} +{"srcContext": "import PFR.TauFunctional\nimport PFR.HundredPercent\nimport PFR.Endgame\n\n/-!\n# Entropic version of polynomial Freiman-Ruzsa conjecture\n\nHere we prove the entropic version of the polynomial Freiman-Ruzsa conjecture.\n\n## Main results\n\n* `entropic_PFR_conjecture`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\n subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 11 d[X^0_1;X^0_2]$.\n\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n\nvariable {\u03a9 \u03a9' : Type*} [m\u03a9 : MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n [MeasurableSingletonClass G]\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G) {X\u2081 : \u03a9 \u2192 G} {X\u2082 : \u03a9 \u2192 G} (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082)\n\n/-- If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that $\\tau[X'_1;X'_2] < \\tau[X_1;X_2]$.\nPhrased in the contrapositive form for convenience of proof. -/\ntheorem tau_strictly_decreases (h_min : tau_minimizes p X\u2081 X\u2082) (hp\u03b7: p.\u03b7 = 1/9): d[X\u2081 # X\u2082] = 0 := by\n let \u27e8A, mA, \u03bc, Y\u2081, Y\u2082, Y\u2081', Y\u2082', h\u03bc, h_indep, hY\u2081, hY\u2082, hY\u2081', hY\u2082', h_id1, h_id2, h_id1', h_id2'\u27e9\n := independent_copies4_nondep hX\u2081 hX\u2082 hX\u2081 hX\u2082 \u2119 \u2119 \u2119 \u2119\n rw [\u2190 h_id1.rdist_eq h_id2]\n letI : MeasureSpace A := \u27e8\u03bc\u27e9\n have : IsProbabilityMeasure (\u2119 : Measure A) := h\u03bc\n rw [\u2190 h_id1.tau_minimizes p h_id2] at h_min\n apply tau_strictly_decreases_aux p Y\u2081 Y\u2082 Y\u2081' Y\u2082' hY\u2081 hY\u2082 hY\u2081' hY\u2082' (h_id1.trans h_id1'.symm)\n (h_id2.trans h_id2'.symm) h_indep h_min hp\u03b7\n\n\n/-- `entropic_PFR_conjecture`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\n subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 11 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture (hp\u03b7 : p.\u03b7 = 1/9):\n \u2203 H : AddSubgroup G, \u2203 \u03a9 : Type uG, \u2203 m\u03a9 : MeasureSpace \u03a9, \u2203 U : \u03a9 \u2192 G,\n IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 Measurable U \u2227\n IsUniform H U \u2227 d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 11 * d[p.X\u2080\u2081 # p.X\u2080\u2082] := by\n obtain \u27e8\u03a9', m\u03a9', X\u2081, X\u2082, hX\u2081, hX\u2082, _, htau_min\u27e9 := tau_minimizer_exists p\n have hdist : d[X\u2081 # X\u2082] = 0 := tau_strictly_decreases p hX\u2081 hX\u2082 htau_min hp\u03b7\n obtain \u27e8H, U, hU, hH_unif, hdistX\u2081, hdistX\u2082\u27e9 := exists_isUniform_of_rdist_eq_zero hX\u2081 hX\u2082 hdist\n refine \u27e8H, \u03a9', inferInstance, U, inferInstance, hU, hH_unif , ?_\u27e9\n have h : \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[p.X\u2080\u2082 # p.X\u2080\u2081 | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1\n rw [tau, tau, hp\u03b7] at h\n norm_num at h\n have : d[p.X\u2080\u2081 # p.X\u2080\u2082] = d[p.X\u2080\u2082 # p.X\u2080\u2081] := rdist_symm\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # X\u2081] + d[X\u2081 # U] := rdist_triangle p.hmeas1 hX\u2081 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # X\u2082] + d[X\u2082 # U] := rdist_triangle p.hmeas2 hX\u2082 hU\n linarith\n\n", "theoremStatement": "theorem entropic_PFR_conjecture' (hp\u03b7 : p.\u03b7 = 1/9):\n \u2203 H : AddSubgroup G, \u2203 \u03a9 : Type uG, \u2203 m\u03a9 : MeasureSpace \u03a9, \u2203 U : \u03a9 \u2192 G,\n IsUniform H U \u2227 d[p.X\u2080\u2081 # U] \u2264 6 * d[p.X\u2080\u2081 # p.X\u2080\u2082] \u2227\n d[p.X\u2080\u2082 # U] \u2264 6 * d[p.X\u2080\u2081 # p.X\u2080\u2082] ", "theoremName": "entropic_PFR_conjecture'", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "89d77f326b0d6ea8e97c7094ce3f7cc22393ee16", "date": "2023-11-18"}, "file": "PFR/PFR/EntropyPFR.lean", "module": "PFR.EntropyPFR", "jsonFile": "PFR.EntropyPFR.jsonl", "positionMetadata": {"lineInFile": 60, "tokenPositionInFile": 2838, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, 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"Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : d[p.X\u2080\u2081 # p.X\u2080\u2082] = d[p.X\u2080\u2082 # p.X\u2080\u2081] := rdist_symm\n obtain \u27e8H, \u03a9, m\u03a9, U, H', hU, hUnif, h'\u27e9 := entropic_PFR_conjecture p hp\u03b7\n refine \u27e8H, \u03a9, m\u03a9, U, hUnif, ?_\u27e9\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # p.X\u2080\u2082] + d[p.X\u2080\u2082 # U] := rdist_triangle p.hmeas1 p.hmeas2 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # p.X\u2080\u2081] + d[p.X\u2080\u2081 # U] := rdist_triangle p.hmeas2 p.hmeas1 hU\n constructor\n \u00b7 linarith\n \u00b7 linarith", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 403}} +{"srcContext": "import PFR.FirstEstimate\nimport PFR.SecondEstimate\nimport PFR.ForMathlib.Elementary\n\n/-!\n# Endgame\n\nThe endgame on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ be independent random variables, with $X_1,\\tilde X_1$ copies\n of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $d[X_1;X_2] = k$\n* $U := X_1 + X_2$\n* $V := \\tilde X_1 + X_2$\n* $W := X_1 + \\tilde X_1$\n* $S := X_1 + X_2 + \\tilde X_1 + \\tilde X_2$.\n* $I_1 := I[U : V | S]$\n* $I_2 := I[U : W | S]$\n* $I_3 := I[V : W | S]$ (not explicitly defined in Lean)\n\n# Main results:\n\n* `sum_condMutual_le` : An upper bound on the total conditional mutual information $I_1+I_2+I_3$.\n* `sum_dist_diff_le`: A sum of the \"costs\" of $U$, $V$, $W$.\n* `construct_good`: A construction of two random variables with small Ruzsa distance between them\n given some random variables with control on total cost, as well as total mutual information.\n-/\n\nopen MeasureTheory ProbabilityTheory\nopen scoped BigOperators\n\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [elem: ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\nvariable {\u03a9 : Type*} [m\u03a9 : MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable (X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G)\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2081', X\u2082'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[U : V | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `V = X\u2081' + X\u2082`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[U : V | S]\n\n/-- `I\u2082 := I[U : W | S]`, the conditional mutual information of `U = X\u2081 + X\u2082` and `W = X\u2081' + X\u2081`\ngiven the quadruple sum `S = X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[U : W | S]\n\n--(Mantas) this times out in the proof below\nprivate lemma hmeas2 :\n Measurable fun p : Fin 4 \u2192 G => ((p 0 + p 1, p 0 + p 2), p 0 + p 1 + p 2 + p 3) := by\n apply Measurable.prod\n \u00b7 apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n\n/-- The quantity $I_3 = I[V:W|S]$ is equal to $I_2$. -/\nlemma I\u2083_eq : I[V : W | S] = I\u2082 := by\n have h_indep2 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081, X\u2082'] := by\n exact h_indep.reindex_four_cbad\n have hident : IdentDistrib (fun a (i : Fin 4) => ![X\u2081, X\u2082, X\u2081', X\u2082'] i a)\n (fun a (j : Fin 4) => ![X\u2081', X\u2082, X\u2081, X\u2082'] j a) := by\n exact { aemeasurable_fst := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n aemeasurable_snd := by\n apply Measurable.aemeasurable\n rw [measurable_pi_iff]\n intro x\n fin_cases x;\n all_goals aesop\n map_eq := by\n rw [\u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081, X\u2082, X\u2081', X\u2082'])\n (Fin.cases hX\u2081 <| Fin.cases hX\u2082 <| Fin.cases hX\u2081' <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep,\n \u2190 (ProbabilityTheory.iIndepFun_iff_pi_map_eq_map (![X\u2081', X\u2082, X\u2081, X\u2082'])\n (Fin.cases hX\u2081' <| Fin.cases hX\u2082 <| Fin.cases hX\u2081 <|\n Fin.cases hX\u2082' Fin.rec0)).mp h_indep2]\n congr\n ext i\n fin_cases i\n all_goals simp [h\u2081.map_eq] }\n have hmeas1 : Measurable (fun p : Fin 4 \u2192 G => (p 0 + p 1, p 0 + p 1 + p 2 + p 3)) := by\n simp_all only [Matrix.cons_val', Matrix.empty_val', Matrix.cons_val_fin_one]\n apply Measurable.prod\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply Measurable.add\n \u00b7 apply Measurable.add\n \u00b7 exact (measurable_pi_apply _).add (measurable_pi_apply _)\n \u00b7 apply measurable_pi_apply\n \u00b7 apply measurable_pi_apply\n have hUVS : IdentDistrib (prod U S) (prod V S) := by\n convert (IdentDistrib.comp hident hmeas1); simp; abel\n have hUVWS : IdentDistrib (prod (prod U W) S) (prod (prod V W) S) := by\n convert (IdentDistrib.comp hident hmeas2) <;> simp <;> abel\n have hU : Measurable U := Measurable.add hX\u2081 hX\u2082\n have hV : Measurable V := Measurable.add hX\u2081' hX\u2082\n have hW : Measurable W := Measurable.add hX\u2081' hX\u2081\n have hS : Measurable S := by measurability\n rw [condMutualInfo_eq hV hW hS, condMutualInfo_eq hU hW hS, chain_rule'' \u2119 hU hS,\n chain_rule'' \u2119 hV hS, chain_rule'' \u2119 hW hS, chain_rule'' \u2119 _ hS, chain_rule'' \u2119 _ hS,\n IdentDistrib.entropy_eq hUVS, IdentDistrib.entropy_eq hUVWS]\n \u00b7 exact Measurable.prod (by exact hU) (by exact hW)\n \u00b7 exact Measurable.prod (by exact hV) (by exact hW)\n\n\n/--\n$$ I(U : V | S) + I(V : W | S) + I(W : U | S) $$\nis less than or equal to\n$$ 6 \\eta k - \\frac{1 - 5 \\eta}{1-\\eta} (2 \\eta k - I_1).$$\n-/\nlemma sum_condMutual_le :\n I[U : V | S] + I[V : W | S] + I[W : U | S]\n \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) := by\n have : I[W:U|S] = I\u2082 := by\n rw [condMutualInfo_comm]\n \u00b7 exact Measurable.add' hX\u2081' hX\u2081\n \u00b7 exact Measurable.add' hX\u2081 hX\u2082\n rw [I\u2083_eq, this]\n have h\u2082 := second_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n have h := add_le_add (add_le_add_left h\u2082 I\u2081) h\u2082\n convert h using 1\n have : 1 - p.\u03b7 > 0 := by linarith [p.h\u03b7']\n field_simp [this]\n ring\n all_goals { simpa }\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" => d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nlemma hU : H[U] = H[X\u2081' + X\u2082'] :=\n IdentDistrib.entropy_eq (ProbabilityTheory.IdentDistrib.add h\u2081 h\u2082\n (iIndepFun.indepFun h_indep (show (0 : Fin 4) \u2260 1 by norm_cast))\n (iIndepFun.indepFun h_indep (show (2 : Fin 4) \u2260 3 by norm_cast)))\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition1 : iIndepFun (fun _ \u21a6 hG) ![X\u2081, X\u2082, X\u2081' + X\u2082'] :=\n h_indep.apply_two_last hX\u2081 hX\u2082 hX\u2081' hX\u2082' measurable_add\n\nlemma hV : H[V] = H[X\u2081 + X\u2082'] :=\nIdentDistrib.entropy_eq (ProbabilityTheory.IdentDistrib.add h\u2081.symm h\u2082\n (iIndepFun.indepFun h_indep (show (2 : Fin 4) \u2260 1 by norm_cast))\n (iIndepFun.indepFun h_indep (show (0 : Fin 4) \u2260 3 by norm_cast)))\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition2 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2081, X\u2081' + X\u2082'] :=\n independenceCondition1 hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_bacd\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition3 : iIndepFun (fun _ \u21a6 hG) ![X\u2081', X\u2082, X\u2081 + X\u2082'] :=\n independenceCondition1 hX\u2081' hX\u2082 hX\u2081 hX\u2082' h_indep.reindex_four_cbad\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition4 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2081', X\u2081 + X\u2082'] :=\n independenceCondition1 hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bcad\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition5 : iIndepFun (fun _ \u21a6 hG) ![X\u2081, X\u2081', X\u2082 + X\u2082'] :=\n independenceCondition1 hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_acbd\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082'} in\nlemma independenceCondition6 : iIndepFun (fun _ \u21a6 hG) ![X\u2082, X\u2082', X\u2081' + X\u2081] :=\n independenceCondition1 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bdca\n\nset_option maxHeartbeats 400000 in\n/--\n$$ \\sum_{i=1}^2 \\sum_{A\\in\\{U,V,W\\}} \\big(d[X^0_i;A|S] - d[X^0_i;X_i]\\big)$$\nis less than or equal to\n$$ \\leq (6 - 3\\eta) k + 3(2 \\eta k - I_1).$$\n-/\nlemma sum_dist_diff_le :\n c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 (6 - 3 * p.\u03b7)*k + 3 * (2*p.\u03b7*k - I\u2081) := by\n let X\u2080\u2081 := p.X\u2080\u2081\n let X\u2080\u2082 := p.X\u2080\u2082\n have ineq1 : d[X\u2080\u2081 # U | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 := by\n have aux1 : H[S] + H[U] - H[X\u2081] - H[X\u2081' + X\u2082'] = H[S] - H[X\u2081] := by\n rw [hU X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep]; ring\n have aux2 : d[X\u2080\u2081 # U | U + (X\u2081' + X\u2082')] - d[X\u2080\u2081 # X\u2081]\n \u2264 (H[U + (X\u2081' + X\u2082')] + H[U] - H[X\u2081] - H[X\u2081' + X\u2082']) / 2 :=\n condRuzsaDist_diff_ofsum_le \u2119 (hX := p.hmeas1) (hY := hX\u2081) (hZ := hX\u2082)\n (Measurable.add hX\u2081' hX\u2082') (independenceCondition1 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n rw [\u2190 add_assoc, aux1] at aux2\n linarith [aux2]\n\n have ineq2 : d[X\u2080\u2082 # U | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := by\n have aux1 : H[S] + H[U] - H[X\u2082] - H[X\u2081' + X\u2082'] = H[S] - H[X\u2082] := by\n rw [hU X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep] ; ring\n have aux2 : d[X\u2080\u2082 # U | U + (X\u2081' + X\u2082')] - d[X\u2080\u2082 # X\u2082]\n \u2264 (H[U + (X\u2081' + X\u2082')] + H[U] - H[X\u2082] - H[X\u2081' + X\u2082']) / 2 := by\n rw [(show U = X\u2082 + X\u2081 from add_comm _ _)]\n apply condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas2) (hX\u2082) (hX\u2081)\n (Measurable.add hX\u2081' hX\u2082') (independenceCondition2 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n rw [\u2190 add_assoc, aux1] at aux2\n linarith [aux2]\n\n have V_add_eq : V + (X\u2081 + X\u2082') = S := by abel\n\n have ineq3 : d[X\u2080\u2081 # V | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 := by\n have aux2 : d[p.X\u2080\u2081 # V | V + (X\u2081 + X\u2082')] - d[p.X\u2080\u2081 # X\u2081']\n \u2264 (H[V + (X\u2081 + X\u2082')] + H[V] - H[X\u2081'] - H[X\u2081 + X\u2082']) / 2 :=\n condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas1) (hX\u2081') (hX\u2082) (Measurable.add hX\u2081 hX\u2082')\n (independenceCondition3 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n\n have aux1 : H[S] + H[V] - H[X\u2081'] - H[X\u2081 + X\u2082'] = H[S ; \u2119] - H[X\u2081 ; \u2119] := by\n rw [hV X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep, h\u2081.entropy_eq]; ring\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl p.hmeas1.aemeasurable) h\u2081,\n V_add_eq, aux1] at aux2\n linarith [aux2]\n\n have ineq4 : d[X\u2080\u2082 # V | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := by\n have aux2 : d[p.X\u2080\u2082 # V | V + (X\u2081 + X\u2082')] - d[p.X\u2080\u2082 # X\u2082]\n \u2264 (H[V + (X\u2081 + X\u2082')] + H[V] - H[X\u2082] - H[X\u2081 + X\u2082']) / 2 := by\n rw [(show V = X\u2082 + X\u2081' from add_comm _ _)]\n apply condRuzsaDist_diff_ofsum_le \u2119 (p.hmeas2) (hX\u2082) (hX\u2081') (Measurable.add hX\u2081 hX\u2082')\n (independenceCondition4 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have aux1 : H[S] + H[V] - H[X\u2082] - H[X\u2081 + X\u2082'] = H[S ; \u2119] - H[X\u2082 ; \u2119] := by\n rw [hV X\u2081 X\u2082 X\u2081' X\u2082' h\u2081 h\u2082 h_indep]; ring\n rw [V_add_eq, aux1] at aux2\n linarith [aux2]\n\n let W' := X\u2082 + X\u2082'\n have ineq5 : d[X\u2080\u2081 # W | S] - d[X\u2080\u2081 # X\u2081] \u2264 (H[S ; \u2119] + H[W ; \u2119] - H[X\u2081 ; \u2119] - H[W' ; \u2119])/2 := by\n have := condRuzsaDist_diff_ofsum_le \u2119 p.hmeas1 hX\u2081 hX\u2081' (Measurable.add hX\u2082 hX\u2082')\n (independenceCondition5 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have S_eq : X\u2081 + X\u2081' + (fun a \u21a6 X\u2082 a + X\u2082' a) = S := by\n rw [(show (fun a \u21a6 X\u2082 a + X\u2082' a) = X\u2082 + X\u2082' by rfl), \u2190 add_assoc, add_assoc X\u2081, add_comm X\u2081',\n \u2190 add_assoc]\n rwa [S_eq, add_comm X\u2081 X\u2081'] at this\n\n have ineq6 : d[X\u2080\u2082 # W' | S] - d[X\u2080\u2082 # X\u2082] \u2264 (H[S ; \u2119] + H[W' ; \u2119] - H[X\u2082 ; \u2119] - H[W ; \u2119])/2 := by\n have := condRuzsaDist_diff_ofsum_le \u2119 p.hmeas2 hX\u2082 hX\u2082' (Measurable.add hX\u2081' hX\u2081)\n (independenceCondition6 hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep)\n have S_eq : X\u2082 + X\u2082' + (fun a \u21a6 X\u2081' a + X\u2081 a) = S := by\n rw [(show (fun a \u21a6 X\u2081' a + X\u2081 a) = X\u2081' + X\u2081 by rfl), add_comm, \u2190 add_assoc, add_comm X\u2081',\n add_assoc X\u2081, add_comm X\u2081', \u2190 add_assoc]\n rwa [S_eq] at this\n\n have dist_eq : d[X\u2080\u2082 # W' | S] = d[X\u2080\u2082 # W | S] := by\n have S_eq : S = (X\u2082 + X\u2082') + (X\u2081' + X\u2081) := by\n rw [add_comm X\u2081' X\u2081, add_assoc _ X\u2082', add_comm X\u2082', \u2190 add_assoc X\u2082, \u2190 add_assoc X\u2082,\n add_comm X\u2082]\n rw [S_eq]\n apply condRuzsaDist'_of_inj_map' p.hmeas2 (hX\u2082.add hX\u2082') (hX\u2081'.add hX\u2081)\n\n -- Put everything together to bound the sum of the `c` terms\n have ineq7 : c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264\n 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := by\n have step\u2081 : c[U|S # U|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc\n _ = (d[p.X\u2080\u2081 # U|S] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2082 # U|S] - d[p.X\u2080\u2082 # X\u2082]) := by ring\n _ \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 + (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := add_le_add ineq1 ineq2\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n have step\u2082 : c[V|S # V|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc c[V|S # V|S] =(d[p.X\u2080\u2081 # V|S] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2082 # V|S] - d[p.X\u2080\u2082 # X\u2082]) := by ring\n _ \u2264 (H[S ; \u2119] - H[X\u2081 ; \u2119])/2 + (H[S ; \u2119] - H[X\u2082 ; \u2119])/2 := add_le_add ineq3 ineq4\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n have step\u2083 : c[W|S # W|S] \u2264 H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 :=\n calc c[W|S # W|S] = (d[X\u2080\u2081 # W | S] - d[X\u2080\u2081 # X\u2081]) + (d[X\u2080\u2082 # W' | S] - d[X\u2080\u2082 # X\u2082]) :=\n by rw [dist_eq]\n _ \u2264 (H[S ; \u2119] + H[W ; \u2119] - H[X\u2081 ; \u2119] - H[W' ; \u2119])/2 + (H[S ; \u2119] + H[W' ; \u2119] - H[X\u2082 ; \u2119] - H[W ; \u2119])/2\n := add_le_add ineq5 ineq6\n _ = H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2 := by ring\n calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) +\n (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) + (H[S ; \u2119] - (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])/2) :=\n add_le_add (add_le_add step\u2081 step\u2082) step\u2083\n _ = 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := by ring\n\n have h_indep' : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'] := by\n apply ProbabilityTheory.iIndepFun.reindex (Equiv.swap (2 : Fin 4) 3)\n convert h_indep using 1\n ext x\n fin_cases x ; all_goals { aesop }\n\n have ineq8 : 3 * H[S ; \u2119] \u2264 3/2 * (H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081 :=\n calc 3 * H[S ; \u2119] \u2264 3 * (H[X\u2081 ; \u2119] / 2 + H[X\u2082 ; \u2119] / 2 + (2+p.\u03b7)*k - I\u2081) := by\n apply (mul_le_mul_left (zero_lt_three' \u211d)).mpr\n (ent_ofsum_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep' h_min)\n _ = 3/2 * ( H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081 := by ring\n\n -- Final computation\n calc c[U|S # U|S] + c[V|S # V|S] + c[W|S # W|S] \u2264 3 * H[S ; \u2119] - 3/2 * H[X\u2081 ; \u2119] -3/2 * H[X\u2082 ; \u2119] := ineq7\n _ = 3 * H[S ; \u2119] - (3/2 *(H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])) := by ring\n _ \u2264 (3/2 * ( H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119]) + 3*(2+p.\u03b7)*k - 3*I\u2081) - (3/2 *(H[X\u2081 ; \u2119] + H[X\u2082 ; \u2119])) :=\n sub_le_sub_right ineq8 _\n _ = (6 - 3 * p.\u03b7)*k + 3 * (2*p.\u03b7*k - I\u2081) := by ring\n\n/-- $U+V+W=0$. -/\nlemma sum_uvw_eq_zero : U+V+W = 0 := by\n rw [add_comm X\u2081' X\u2082, ElementaryAddCommGroup.sum_add_sum_add_sum_eq_zero]\n\nsection construct_good\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081+T\u2082+T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\nThen there exist random variables $T'_1, T'_2$ such that\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X_1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2]) $$\nis at most\n$$ \\delta + \\eta ( d[X^0_1;T_1]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2]-d[X^0_2;X_2]) $$\n$$ + \\tfrac12 \\eta I[T_1: T_3] + \\tfrac12 \\eta I[T_2: T_3].$$\n-/\nlemma construct_good_prelim :\n k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 # T\u2082] + p.\u03b7 * (I[T\u2081: T\u2083] + I[T\u2082 : T\u2083])/2 := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n\n have hp.\u03b7 : 0 \u2264 p.\u03b7 := by linarith [p.h\u03b7]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 :=\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have h2T\u2081 : T\u2081 = T\u2082 + T\u2083 := by simp [h2T\u2083, add_left_comm]\n have h2T\u2082 : T\u2082 = T\u2083 + T\u2081 := by simp [h2T\u2081, add_left_comm]\n\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n\n have h2 : p.\u03b7 * sum2 \u2264 p.\u03b7 * (d[p.X\u2080\u2081 # T\u2081] - d[p.X\u2080\u2081 # X\u2081] + I[T\u2081 : T\u2083] / 2) := by\n have : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), sum2]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n gcongr\n linarith [condRuzsaDist_le' \u2119 \u2119 p.hmeas1 hT\u2081 hT\u2083]\n\n have h3 : p.\u03b7 * sum3 \u2264 p.\u03b7 * (d[p.X\u2080\u2082 # T\u2082] - d[p.X\u2080\u2082 # X\u2082] + I[T\u2082 : T\u2083] / 2) := by\n have : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp [sum3, integral_sub (.of_finite _ _) (.of_finite _ _)]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n gcongr\n linarith [condRuzsaDist_le' \u2119 \u2119 p.hmeas2 hT\u2082 hT\u2083]\n\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * c[T\u2081 # T\u2082] + p.\u03b7 * (I[T\u2081 : T\u2083] + I[T\u2082 : T\u2083]) / 2 := by\n suffices sum4 = sum1 + p.\u03b7 * (sum2 + sum3) by linarith\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _), integral_mul_left]\n\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $ ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\n", "theoremStatement": "/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n-\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\n\nThen there exist random variables $T'_1, T'_2$ such that\n\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X _1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2])$$\n\nis at most\n\n$$\\delta + \\frac{\\eta}{3} \\biggl( \\delta + \\sum_{i=1}^2 \\sum_{j = 1}^3\n (d[X^0_i;T_j] - d[X^0_i; X_i]) \\biggr).$$\n-/\nlemma construct_good : k \u2264 \u03b4 + (p.\u03b7/3) * (\u03b4 + c[T\u2081 # T\u2081] + c[T\u2082 # T\u2082] + c[T\u2083 # T\u2083]) ", "theoremName": "construct_good", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "ab757c1077e01bb24ff09343c0f7775b88174d12", "date": "2023-11-16"}, "file": "PFR/PFR/Endgame.lean", "module": "PFR.Endgame", "jsonFile": "PFR.Endgame.jsonl", "positionMetadata": {"lineInFile": 408, "tokenPositionInFile": 18648, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 248, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", 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"Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have v2 := construct_good_prelim p X\u2081 X\u2082 h_min (by rw [\u2190 hT]; abel) hT\u2081 hT\u2083 hT\u2082\n have v3 := construct_good_prelim p X\u2081 X\u2082 h_min (by rw [\u2190 hT]; abel) hT\u2082 hT\u2081 hT\u2083\n have v6 := construct_good_prelim p X\u2081 X\u2082 h_min (by rw [\u2190 hT]; abel) hT\u2083 hT\u2082 hT\u2081\n simp only [mutualInfo, entropy_comm hT\u2082 hT\u2081, entropy_comm hT\u2083 hT\u2081, entropy_comm hT\u2083 hT\u2082]\n at *\n linarith", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 362}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] := by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]\n\n/-- Adding a constant to a random variable does not change the Rusza distance. -/\nlemma rdist_add_const [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y + fun _ \u21a6 c; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n have A : IdentDistrib (Y' + fun _ \u21a6 c) (Y + fun _ \u21a6 c) \u03bd \u03bc' := by\n change IdentDistrib (fun \u03c9 \u21a6 Y' \u03c9 + c) (fun \u03c9 \u21a6 Y \u03c9 + c) \u03bd \u03bc'\n apply hIdY.comp (measurable_add_const c)\n have B : IndepFun X' (Y' + fun _ \u21a6 c) \u03bd := by\n change IndepFun X' (fun \u03c9 \u21a6 Y' \u03c9 + c) \u03bd\n apply hind.comp measurable_id (measurable_add_const c)\n have C : X' - (Y' + fun _ \u21a6 c) = (X' - Y') + (fun _ \u21a6 -c) := by\n ext \u03c9; simp; abel\n rw [\u2190 hIdX.rdist_eq hIdY, \u2190 hIdX.rdist_eq A, hind.rdist_eq hX' hY',\n B.rdist_eq hX' (hY'.add_const _), entropy_add_const hY' c, C, entropy_add_const]\n exact hX'.sub hY'\n\n/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\nlemma rdist_add_const' [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] (c: G) (c': G)\n (hX : Measurable X) (hY : Measurable Y) :\n d[X + fun _ \u21a6 c; \u03bc # Y + fun _ \u21a6 c'; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_add_const _ hY, rdist_symm, rdist_add_const hY hX, rdist_symm]\n measurability\n\n/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (X : \u03a9 \u2192 G) (Y : \u03a9 \u2192 G) (Z : \u03a9 \u2192 G)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n have h1 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] + H[X - Y; \u03bc] \u2264 H[\u27e8X - Z, X - Y\u27e9; \u03bc] + H[\u27e8Y, X - Y\u27e9; \u03bc] :=\n entropy_triple_add_entropy_le \u03bc (hX.sub hZ) hY (hX.sub hY)\n have h2 : H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n calc H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n have : \u27e8X - Z, X - Y\u27e9 = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 \u27e8X - Z, Y - Z\u27e9 := by ext1; simp\n rw [this]\n apply entropy_comp_le \u03bc (by measurability)\n _ \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n have h : 0 \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] - H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n apply mutualInfo_nonneg (by measurability) (by measurability) \u03bc\n linarith\n have h3 : H[\u27e8Y, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X, Y\u27e9 ; \u03bc] := by\n have : \u27e8Y, X - Y\u27e9 = (fun p \u21a6 (p.2, p.1 - p.2)) \u2218 \u27e8X, Y\u27e9 := by ext1; simp\n rw [this]\n exact entropy_comp_le \u03bc (hX.prod_mk hY) _\n have h4 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] = H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] := by\n refine entropy_of_comp_eq_of_comp \u03bc ((hX.sub hZ).prod_mk (hY.prod_mk (hX.sub hY)))\n (hX.prod_mk (hY.prod_mk hZ))\n (fun p : G \u00d7 (G \u00d7 G) \u21a6 (p.2.2 + p.2.1, p.2.1, -p.1 + p.2.2 + p.2.1))\n (fun p : G \u00d7 G \u00d7 G \u21a6 (p.1 - p.2.2, p.2.1, p.1 - p.2.1)) ?_ ?_\n \u00b7 ext1; simp\n \u00b7 ext1; simp\n have h5 : H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [entropy_assoc hX hY hZ, entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h\n rw [h4, h5] at h1\n calc H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Y - Z; \u03bc] - H[Z; \u03bc] := by linarith\n _ = H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n congr 2\n rw [entropy_sub_comm hY hZ]\n\n/-- The **entropic Ruzsa triangle inequality** -/\nlemma rdist_triangle {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [h\u03bc : IsProbabilityMeasure \u03bc] [h\u03bc' : IsProbabilityMeasure \u03bc'] [h\u03bc'' : IsProbabilityMeasure \u03bc''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n d[X ; \u03bc # Z ; \u03bc''] \u2264 d[X ; \u03bc # Y ; \u03bc'] + d[Y ; \u03bc' # Z ; \u03bc''] := by\n obtain \u27e8A, mA, \u03bcA, X', Y', Z', h\u03bcA, hInd, hX', hY', hZ', HX, HY, HZ, _, _, _\u27e9 :=\n independent_copies3_nondep_finiteRange hX hY hZ \u03bc \u03bc' \u03bc''\n suffices d[X' ; \u03bcA # Z' ; \u03bcA] \u2264 d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] by\n rwa [HX.rdist_eq HY, HY.rdist_eq HZ, HX.rdist_eq HZ] at this\n have IndepLem : IndepFun (\u27e8X', Z'\u27e9) Y' \u03bcA :=\n iIndepFun.indepFun_prod_mk hInd (fun i => by fin_cases i ; all_goals { simpa }) 0 2 1\n (by norm_cast) (by norm_cast)\n calc d[X' ; \u03bcA # Z' ; \u03bcA] = H[X' - Z'; \u03bcA] - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) := by\n rw [ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using hInd.indepFun (show 0 \u2260 2 by norm_cast)) hX' hZ'] ; ring\n _ \u2264 (H[X' - Y' ; \u03bcA] + H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA]) - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) :=\n sub_le_sub_right (ent_of_diff_le _ _ _ hX' hZ' hY' IndepLem) _\n _ = (H[X' - Y' ; \u03bcA] - H[X'; \u03bcA] / 2 - H[Y' ; \u03bcA] / 2) +\n (H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA] / 2 - H[Z'; \u03bcA] / 2) := by ring\n _ = d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] := by\n rw [ProbabilityTheory.IndepFun.rdist_eq (by simpa using (iIndepFun.indepFun hInd\n (show 0 \u2260 1 by norm_cast))) hX' hY', ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using (iIndepFun.indepFun hInd (show 1 \u2260 2 by norm_cast))) hY' hZ']\n\nvariable [MeasurableSingletonClass S] [MeasurableSingletonClass T]\n\n/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) [IsFiniteMeasure \u03bc]\n (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc'\"]\" => condRuzsaDist X Z Y W \u03bc \u03bc'\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" # \" Y \" | \" W \"]\" => condRuzsaDist X Z Y W volume volume\n\nlemma condRuzsaDist_def (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W] := rfl\n\n/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\nlemma condRuzsaDist_symm {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hZ : Measurable Z)\n (hW : Measurable W) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange Z]\n [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[Y | W ; \u03bc' # X | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist_symm]\n\n@[simp] lemma condRuszaDist_zero_right (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n d[X | Z ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero,\n kernel.rdist_zero_right]\n\n@[simp] lemma condRuszaDist_zero_left (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; 0 # Y | W ; \u03bc'] = 0 := by\n simp [condRuzsaDist]\n\nlemma condRuzsaDist_nonneg {X : \u03a9 \u2192 G} (hX : Measurable X) [FiniteRange X]\n {Z : \u03a9 \u2192 S} (hZ : Measurable Z) [FiniteRange Z]\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) [FiniteRange Y]\n {W : \u03a9' \u2192 T} (hW : Measurable W) [FiniteRange W]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n 0 \u2264 d[X | Z ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def]\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n refine kernel.rdist_nonneg ?_ ?_\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hX hZ\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hY hW\n\n/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\nlemma rdist_eq_rdistm : d[X ; \u03bc # Y ; \u03bc'] = kernel.rdistm (\u03bc.map X) (\u03bc'.map Y) := rfl\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\nlemma condRuzsaDist_eq_sum {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z in FiniteRange.toFinset Z, \u2211 w in FiniteRange.toFinset W,\n (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((((FiniteRange.toFinset Z)\n \u00d7\u02e2 (FiniteRange.toFinset W)) : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Finset.sum_product,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\nlemma condRuzsaDist_eq_sum' {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [Fintype S] [Fintype T] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z, \u2211 w, (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Fintype.sum_prod_type,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist' (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc' \"]\" => condRuzsaDist' X Y W \u03bc \u03bc'\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" # \" Y \" | \" W \"]\" => condRuzsaDist' X Y W volume volume\n\n/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\nlemma condRuzsaDist'_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y | W ; \u03bc'] =\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W] :=\n rfl\n\n@[simp] lemma condRuzsaDist'_zero_right (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) :\n d[X ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist'_def, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero, kernel.rdist_zero_right]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\nlemma condRuzsaDist'_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (dirac ()) (\u03bc'.map W) ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 FiniteRange.toFinset W : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, smul_eq_mul, Finset.sum_product]\n simp only [Finset.univ_unique, PUnit.default_eq_unit, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, Pi.one_apply, one_mul,\n Finset.sum_singleton]\n simp_rw [Measure.map_apply hW (measurableSet_singleton _)]\n congr with w\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff]\n refine Or.inr ?_\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm, condDistrib_apply hY hW _ _ hw]\n congr\n\n/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\nlemma condRuzsaDist'_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_eq_sum hY hW \u03bc \u03bc']\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\nopen scoped ENNReal\n\nlemma condRuzsaDist'_prod_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [FiniteRange W] [FiniteRange W']:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n have : d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc'] = \u2211 w in ((FiniteRange.toFinset W') \u00d7\u02e2 FiniteRange.toFinset W), (\u03bc' ((fun a => (W' a, W a)) \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; \u03bc'[|(fun a => (W' a, W a)) \u207b\u00b9' {w}]] := by\n rw [condRuzsaDist'_eq_sum hY (hW'.prod_mk hW)]\n apply Finset.sum_subset\n . intro (t, t')\n simp\n intro \u03c9 h1 h2\n exact \u27e8\u27e8\u03c9, h1\u27e9, \u27e8\u03c9, h2\u27e9\u27e9\n intro (t, t') _ ht\n simp at ht\n have : (fun a => (W' a, W a)) \u207b\u00b9' {(t, t')} = \u2205 := by\n ext \u03c9\n simp\n exact ht \u03c9\n simp [this]\n rw [this, Finset.sum_product_right]\n congr 1 with w\n rw [condRuzsaDist'_eq_sum hY hW', Finset.mul_sum]\n congr 1 with w'\n rw [\u2190 mul_assoc]\n have A : (fun a \u21a6 (W' a, W a)) \u207b\u00b9' {(w', w)} = W' \u207b\u00b9' {w'} \u2229 W\u207b\u00b9' {w} := by ext; simp\n congr 1\n \u00b7 simp only [ProbabilityTheory.cond, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n smul_eq_mul, ENNReal.toReal_mul, A, restrict_apply (hW' (MeasurableSet.singleton w'))]\n rcases eq_bot_or_bot_lt (\u03bc' (W \u207b\u00b9' {w})) with hw|hw\n \u00b7 have : \u03bc' (W' \u207b\u00b9' {w'} \u2229 W \u207b\u00b9' {w}) = 0 :=\n le_antisymm (le_trans (measure_mono (Set.inter_subset_right _ _)) hw.le) bot_le\n simp [hw, this]\n \u00b7 rw [\u2190 mul_assoc, \u2190 ENNReal.toReal_mul, ENNReal.mul_inv_cancel, ENNReal.one_toReal, one_mul]\n exacts [hw.ne', by finiteness]\n \u00b7 congr 1\n rw [A, cond_cond_eq_cond_inter' _ (hW (.singleton w)) (hW' (.singleton w')), Set.inter_comm]\n finiteness\n\n/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\nlemma condRuzsaDist'_prod_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [Fintype T]:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_prod_eq_sum \u03bc \u03bc' hY hW' hW]\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\nlemma condRuzsaDist'_eq_integral (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = (\u03bc'.map W)[fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])]] := by\n rw [condRuzsaDist'_eq_sum hY hW]\n simp_rw [\u2190 smul_eq_mul]\n have : (\u03bc'.map W) (FiniteRange.toFinset W : Set T)\u1d9c = 0 := by\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n convert symm $ integral_eq_sum' (\u03bc'.map W) this (fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])])\n rw [Measure.map_apply hW (MeasurableSet.singleton _)]\n\n/-- Conditioning by a constant does not affect Ruzsa distance. -/\nlemma condRuzsaDist_of_const {X : \u03a9 \u2192 G} (hX : Measurable X) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (c : S)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange W] :\n d[X|(fun _ \u21a6 c) ; \u03bc # Y | W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def, condRuzsaDist'_def, Measure.map_const,measure_univ,one_smul, kernel.rdist,\n kernel.rdist, integral_prod, integral_dirac, integral_prod,integral_dirac]\n dsimp; congr; ext x; congr\n rw [condDistrib_apply hX measurable_const]\n \u00b7 simp\n \u00b7 simp\n \u00b7 exact integrable_of_finiteSupport _\n \u00b7 exact integrable_of_finiteSupport _\n\n/-- If $(X,Z)$ and $(Y,W)$ are independent, then\n$$ d[X | Z ; Y | W] = H[X'- Y'|Z', W'] - H[X'|Z']/2 - H[Y'|W']/2$$\n-/\nlemma condRuzsaDist_of_indep\n {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun (\u27e8X, Z\u27e9) (\u27e8Y, W\u27e9) \u03bc) [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc] = H[X - Y | \u27e8Z, W\u27e9 ; \u03bc] - H[X | Z ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ (hZ.prod_mk hW),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hW]\n swap; \u00b7 exact hX.sub hY\n congr 2\n have hZW : IndepFun Z W \u03bc := by\n have h' := IndepFun.comp h measurable_snd measurable_snd\n exact h'\n have hZW_map : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map Z).prod (\u03bc.map W) :=\n (indepFun_iff_map_prod_eq_prod_map_map hZ.aemeasurable hW.aemeasurable).mp hZW\n rw [\u2190 hZW_map]\n refine kernel.entropy_congr ?_\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (hZ.prod_mk hW) _ _).symm\n refine (this.symm.trans ?_).symm\n suffices kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft S (condDistrib Y W \u03bc)\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc by\n filter_upwards [this] with x hx\n rw [kernel.map_apply, kernel.map_apply, hx]\n . exact (condDistrib_eq_prod_of_indepFun hX hZ hY hW \u03bc h).symm\n\n/-- Formula for conditional Ruzsa distance for independent sets of variables. -/\nlemma condRuzsaDist'_of_indep {X : \u03a9 \u2192 G} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun X (\u27e8Y, W\u27e9) \u03bc) [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc] = H[X - Y | W ; \u03bc] - H[X ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist'_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ hW,\n condEntropy_eq_kernel_entropy hY hW, entropy_eq_kernel_entropy]\n rotate_left\n \u00b7 exact hX.sub hY\n congr 2\n let Z : \u03a9 \u2192 Unit := fun _ \u21a6 ()\n rw [\u2190 condDistrib_unit_right hX \u03bc]\n have h' : IndepFun (\u27e8X,Z\u27e9) (\u27e8Y, W\u27e9) \u03bc := by\n rw [indepFun_iff_measure_inter_preimage_eq_mul]\n intro s t hs ht\n have : \u27e8X, Z\u27e9 \u207b\u00b9' s = X \u207b\u00b9' ((fun c \u21a6 (c, ())) \u207b\u00b9' s) := by ext1 y; simp\n rw [this]\n rw [indepFun_iff_measure_inter_preimage_eq_mul] at h\n exact h _ _ (measurable_prod_mk_right hs) ht\n have h_indep := condDistrib_eq_prod_of_indepFun hX measurable_const hY hW _ h'\n have h_meas_eq : \u03bc.map (\u27e8Z, W\u27e9) = (Measure.dirac ()).prod (\u03bc.map W) := by\n ext s hs\n rw [Measure.map_apply (measurable_const.prod_mk hW) hs, Measure.prod_apply hs, lintegral_dirac,\n Measure.map_apply hW (measurable_prod_mk_left hs)]\n congr\n rw [\u2190 h_meas_eq]\n have : kernel.map (kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft Unit (condDistrib Y W \u03bc)) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc)\n (fun x \u21a6 x.1 - x.2) measurable_sub := by\n filter_upwards [h_indep] with y hy\n conv_rhs => rw [kernel.map_apply, hy]\n rw [kernel.entropy_congr this]\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (measurable_const.prod_mk hW) _ _).symm\n rw [kernel.entropy_congr this]\n have h_meas : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map W).map (Prod.mk ()) := by\n ext s hs\n rw [Measure.map_apply measurable_prod_mk_left hs, h_meas_eq, Measure.prod_apply hs,\n lintegral_dirac]\n have h_ker : condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.prodMkLeft Unit (condDistrib (X - Y) W \u03bc) := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply (measurable_const.prod_mk hW) (measurableSet_singleton _)] at hx\n ext s hs\n have h_preimage_eq : (fun a \u21a6 (PUnit.unit, W a)) \u207b\u00b9' {x} = W \u207b\u00b9' {x.2} := by\n conv_lhs => rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod]\n ext1 y\n simp\n rw [kernel.prodMkLeft_apply, condDistrib_apply' _ (measurable_const.prod_mk hW) _ _ hx hs,\n condDistrib_apply' _ hW _ _ _ hs]\n rotate_left\n \u00b7 exact hX.sub hY\n \u00b7 convert hx\n exact h_preimage_eq.symm\n \u00b7 exact hX.sub hY\n congr\n rw [kernel.entropy_congr h_ker, h_meas, kernel.entropy_prodMkLeft_unit]\n\n", "theoremStatement": "/-- The conditional Ruzsa distance is unchanged if the sets of random variables are replaced with\ncopies. -/\nlemma condRuzsaDist_of_copy {X : \u03a9 \u2192 G} (hX : Measurable X) {Z : \u03a9 \u2192 S} (hZ : Measurable Z)\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) {W : \u03a9' \u2192 T} (hW : Measurable W)\n {X' : \u03a9'' \u2192 G} (hX' : Measurable X') {Z' : \u03a9'' \u2192 S} (hZ' : Measurable Z')\n {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib (\u27e8X, Z\u27e9) (\u27e8X', Z'\u27e9) \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange Z] [FiniteRange W] [FiniteRange Z'] [FiniteRange W'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[X' | Z' ; \u03bc'' # Y' | W' ; \u03bc'''] ", "theoremName": "condRuzsaDist_of_copy", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc621d65f3d5b43306793549b7af0b5c922d3", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 776, "tokenPositionInFile": 39463, "theoremPositionInFile": 49}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 13, "numPremises": 201, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", 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"Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "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.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n classical\n set A := (FiniteRange.toFinset Z) \u222a (FiniteRange.toFinset Z')\n set B := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n have hfull' : Measure.prod (\u03bc''.map Z') (\u03bc'''.map W') ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hZZ' : \u03bc.map Z = \u03bc''.map Z' := (h1.comp measurable_snd).map_eq\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hZZ', \u2190 hWW',\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hz : \u03bc (Z \u207b\u00b9' {x.1}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 have hZZ'x : \u03bc (Z \u207b\u00b9' {x.1}) = \u03bc'' (Z' \u207b\u00b9' {x.1}) := by\n have : \u03bc.map Z {x.1} = \u03bc''.map Z' {x.1} := by rw [hZZ']\n rwa [Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hZ' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hX hZ _ _ hz hs, condDistrib_apply' hX' hZ' _ _ _ hs]\n swap; \u00b7 rwa [hZZ'x] at hz\n congr\n have : \u03bc.map (\u27e8X, Z\u27e9) (s \u00d7\u02e2 {x.1}) = \u03bc''.map (\u27e8X', Z'\u27e9) (s \u00d7\u02e2 {x.1}) := by rw [h1.map_eq]\n rwa [Measure.map_apply (hX.prod_mk hZ) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hX'.prod_mk hZ') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 X' a) \u207b\u00b9' s)] at this\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this", "proofType": "tactic", "proofLengthLines": 64, "proofLengthTokens": 3140}}