module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.DedekindDomain.PID | {
"line": 66,
"column": 49
} | {
"line": 83,
"column": 85
} | {
"line": 85,
"column": 0
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nh : Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v)) = ⊤\n⊢ (↑↑I).IsPrincipal",
"ppTerm": "?m.61",
"a... | [] | by
have hinv := I.mul_inv
set J := Submodule.comap (Algebra.linearMap R A) ((I : Submodule R A) * Submodule.span R {v})
have hJ : IsLocalization.coeSubmodule A J = ↑I * Submodule.span R {v} := by
rw [coe_ext_iff, coe_mul, coe_one] at hinv
apply Submodule.map_comap_eq_self
grw [← Submodule.one_eq_range... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.PID | {
"line": 121,
"column": 12
} | {
"line": 121,
"column": 14
} | {
"line": 121,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nhS : S ≤ R⁰\nhf : {I | I.IsMaximal}.Finite\nI I' : FractionalIdeal S A\nhinv : ↑I * ↑I' = ↑1\nhinv' : I * I' = 1\ns : Finset (Ideal R) := hf.toFinset\nthis : Decidabl... | [
"R : Type u_1\ninst✝³ : CommRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nhS : S ≤ R⁰\nhf : {I | I.IsMaximal}.Finite\nI I' : FractionalIdeal S A\nhinv : ↑I * ↑I' = ↑1\nhinv' : I * I' = 1\ns : Finset (Ideal R) := hf.toFinset\nthis : DecidableEq (Ideal R... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Ideal.Int | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 31
} | {
"line": 126,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\ninst✝² : Algebra.IsIntegral ℤ R\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : NeZero P\n⊢ ¬under ℤ P = ⊥",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"Ne",
... | [
"R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\ninst✝² : Algebra.IsIntegral ℤ R\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : NeZero P\nthis : P ≠ ⊥\n⊢ ¬under ℤ P = ⊥"
] | have : P ≠ ⊥ := NeZero.ne _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Instances | {
"line": 136,
"column": 25
} | {
"line": 136,
"column": 63
} | {
"line": 136,
"column": 63
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : FaithfulSMul R S\nx : R\n⊢ ((algebraMap Rₚ L).comp (algebraMap R Rₚ)) x ... | [] | by simp [RingHom.algebraMap_toAlgebra] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.Instances | {
"line": 139,
"column": 41
} | {
"line": 139,
"column": 79
} | {
"line": 139,
"column": 79
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : FaithfulSMul R S\nx : R\n⊢ (algebraMap R K) x = ((algebraMap Rₚ K).comp ... | [] | by simp [RingHom.algebraMap_toAlgebra] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 70
} | {
"line": 116,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝²⁴ : CommRing R\ninst✝²³ : IsDomain R\nS : Type u_3\ninst✝²² : CommRing S\ninst✝²¹ : IsDomain S\ninst✝²⁰ : IsIntegrallyClosed R\ninst✝¹⁹ : IsIntegrallyClosed S\ninst✝¹⁸ : Algebra R S\ninst✝¹⁷ : Module.Finite R S\ninst✝¹⁶ : IsTorsionFree R S\nI : Ideal S\nM : Submonoid R\nhM : M ≤ R⁰\... | [
"R : Type u_1\ninst✝²⁴ : CommRing R\ninst✝²³ : IsDomain R\nS : Type u_3\ninst✝²² : CommRing S\ninst✝²¹ : IsDomain S\ninst✝²⁰ : IsIntegrallyClosed R\ninst✝¹⁹ : IsIntegrallyClosed S\ninst✝¹⁸ : Algebra R S\ninst✝¹⁷ : Module.Finite R S\ninst✝¹⁶ : IsTorsionFree R S\nI : Ideal S\nM : Submonoid R\nhM : M ≤ R⁰\nRₘ : Type u... | let f : Rₘ →+* K := IsLocalization.map _ (T := R⁰) (RingHom.id R) hM | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 157,
"column": 12
} | {
"line": 157,
"column": 14
} | {
"line": 158,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝²⁴ : CommRing R\ninst✝²³ : IsDomain R\nS : Type u_3\ninst✝²² : CommRing S\ninst✝²¹ : IsDomain S\ninst✝²⁰ : IsIntegrallyClosed R\ninst✝¹⁹ : IsIntegrallyClosed S\ninst✝¹⁸ : Algebra R S\ninst✝¹⁷ : Module.Finite R S\ninst✝¹⁶ : IsTorsionFree R S\nI : Ideal S\nM : Submonoid ... | [
"case refine_2\nR : Type u_1\ninst✝²⁴ : CommRing R\ninst✝²³ : IsDomain R\nS : Type u_3\ninst✝²² : CommRing S\ninst✝²¹ : IsDomain S\ninst✝²⁰ : IsIntegrallyClosed R\ninst✝¹⁹ : IsIntegrallyClosed S\ninst✝¹⁸ : Algebra R S\ninst✝¹⁷ : Module.Finite R S\ninst✝¹⁶ : IsTorsionFree R S\nI : Ideal S\nM : Submonoid R\nhM : M ≤ ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 14
} | {
"line": 190,
"column": 6
} | [
{
"pp": "case mem\nR : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\nS : Type u_3\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : IsIntegrallyClosed R\ninst✝³ : IsIntegrallyClosed S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\ninst✝ : IsTorsionFree R S\nI : Ideal S\nx x✝ : R\nh : x✝ ∈ ⇑(Algebra.... | [] | | mem _ h => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 15
} | {
"line": 97,
"column": 2
} | [
{
"pp": "case refine_1\np k : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nhcycl : IsCyclotomicExtension {p ^ k} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ k)\nx : K\nh : IsIntegral ℤ x\nB : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ\nhint : IsIntegral ℤ B.gen\nthis : ... | [
"case refine_1\np k : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nhcycl : IsCyclotomicExtension {p ^ k} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ k)\nx : K\nh : IsIntegral ℤ x\nB : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ\nhint : IsIntegral ℤ B.gen\nthis : FiniteDimens... | rw [hun] at H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 201,
"column": 75
} | {
"line": 201,
"column": 88
} | {
"line": 201,
"column": 89
} | [
{
"pp": "R : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _r... | [
"R : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _root_.Prime p... | mul_comm _ p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 592,
"column": 10
} | {
"line": 592,
"column": 49
} | {
"line": 593,
"column": 8
} | [
{
"pp": "case neg\nn : ℕ\nhn✝ : 2 ≤ n\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : K\np : ℕ\nhF : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ n\nthis : NeZero n\nμ : ↥ℚ⟮ζ⟯\nhC : IsCyclotomicExtension {n} ℚ ↥ℚ⟮ζ⟯\nhμ : IsPrimitiveRoot μ n\nhp : p = 1\nh✝ : (Algebra.norm ℤ) (hζ.toInteger - 1) = (Alg... | [] | exact (Nat.Prime.ne_one hF.out hp).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 592,
"column": 10
} | {
"line": 592,
"column": 49
} | {
"line": 593,
"column": 8
} | [
{
"pp": "case neg\nn : ℕ\nhn✝ : 2 ≤ n\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : K\np : ℕ\nhF : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ n\nthis : NeZero n\nμ : ↥ℚ⟮ζ⟯\nhC : IsCyclotomicExtension {n} ℚ ↥ℚ⟮ζ⟯\nhμ : IsPrimitiveRoot μ n\nhp : p = 1\nh✝ : (Algebra.norm ℤ) (hζ.toInteger - 1) = (Alg... | [] | exact (Nat.Prime.ne_one hF.out hp).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 592,
"column": 10
} | {
"line": 592,
"column": 49
} | {
"line": 593,
"column": 8
} | [
{
"pp": "case neg\nn : ℕ\nhn✝ : 2 ≤ n\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : K\np : ℕ\nhF : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ n\nthis : NeZero n\nμ : ↥ℚ⟮ζ⟯\nhC : IsCyclotomicExtension {n} ℚ ↥ℚ⟮ζ⟯\nhμ : IsPrimitiveRoot μ n\nhp : p = 1\nh✝ : (Algebra.norm ℤ) (hζ.toInteger - 1) = (Alg... | [] | exact (Nat.Prime.ne_one hF.out hp).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 297,
"column": 10
} | {
"line": 297,
"column": 12
} | {
"line": 298,
"column": 2
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : Field K\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra A K\ninst✝¹³ : Algebra B L\ninst✝¹² : Algebra A B\ninst✝¹¹ : Algebra K L\ninst✝¹⁰ : Algebra A L\ninst✝⁹ : IsScalarTower A K L\ninst✝⁸ : IsScalarTower... | [
"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : Field K\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra A K\ninst✝¹³ : Algebra B L\ninst✝¹² : Algebra A B\ninst✝¹¹ : Algebra K L\ninst✝¹⁰ : Algebra A L\ninst✝⁹ : IsScalarTower A K L\ninst✝⁸ : IsScalarTower A B L\ninst... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 337,
"column": 2
} | {
"line": 339,
"column": 87
} | {
"line": 341,
"column": 0
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : Field K\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra A K\ninst✝¹³ : Algebra B L\ninst✝¹² : Algebra A B\ninst✝¹¹ : Algebra K L\ninst✝¹⁰ : Algebra A L\ninst✝⁹ : IsScalarTower A K L\ninst✝⁸ : IsScalarTower... | [] | by_cases hI : I = 0
· simp [hI]
rw [← coe_le_coe, ← coe_le_coe, coe_mul, coe_dual A K hJ, coe_dual_one, le_traceDual] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 337,
"column": 2
} | {
"line": 339,
"column": 87
} | {
"line": 341,
"column": 0
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : Field K\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra A K\ninst✝¹³ : Algebra B L\ninst✝¹² : Algebra A B\ninst✝¹¹ : Algebra K L\ninst✝¹⁰ : Algebra A L\ninst✝⁹ : IsScalarTower A K L\ninst✝⁸ : IsScalarTower... | [] | by_cases hI : I = 0
· simp [hI]
rw [← coe_le_coe, ← coe_le_coe, coe_mul, coe_dual A K hJ, coe_dual_one, le_traceDual] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 614,
"column": 2
} | {
"line": 615,
"column": 29
} | {
"line": 616,
"column": 2
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : Algebra K L\ninst✝⁶ : Algebra A L\ninst✝⁵ : IsScalarTower A K L\ninst✝⁴ : IsDomain A\ninst✝³ : IsFractionRing A K\ninst✝² : FiniteDimensional K L\ninst✝¹ : Algebra.IsSeparab... | [
"A : Type u_1\nK : Type u_2\nL : Type u\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : Algebra K L\ninst✝⁶ : Algebra A L\ninst✝⁵ : IsScalarTower A K L\ninst✝⁴ : IsDomain A\ninst✝³ : IsFractionRing A K\ninst✝² : FiniteDimensional K L\ninst✝¹ : Algebra.IsSeparable K L\ninst... | rw [← span_coeff_minpolyDiv hAx, LinearMap.BilinForm.dualSubmodule_span_of_basis _ hnondeg,
Submodule.smul_span, hpb] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 764,
"column": 4
} | {
"line": 764,
"column": 61
} | {
"line": 766,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : NumberField K\nF₁ F₂ : IntermediateField ℚ K\nn₁ n₂ : ℕ\ninst✝³ : NeZero n₁\ninst✝² : NeZero n₂\ninst✝¹ : IsCyclotomicExtension {n₁} ℚ ↥F₁\ninst✝ : IsCyclotomicExtension {n₂} ℚ ↥F₂\nζ₁ : ↥F₁\nhζ₁✝ : IsPrimitiveRoot ζ₁ n₁\nh₁ : ℤ[hζ₁✝.toInteger... | [] | exact isCoprime_differentIdeal_of_isCoprime_discr _ h_cpr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 764,
"column": 4
} | {
"line": 764,
"column": 61
} | {
"line": 766,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : NumberField K\nF₁ F₂ : IntermediateField ℚ K\nn₁ n₂ : ℕ\ninst✝³ : NeZero n₁\ninst✝² : NeZero n₂\ninst✝¹ : IsCyclotomicExtension {n₁} ℚ ↥F₁\ninst✝ : IsCyclotomicExtension {n₂} ℚ ↥F₂\nζ₁ : ↥F₁\nhζ₁✝ : IsPrimitiveRoot ζ₁ n₁\nh₁ : ℤ[hζ₁✝.toInteger... | [] | exact isCoprime_differentIdeal_of_isCoprime_discr _ h_cpr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 764,
"column": 4
} | {
"line": 764,
"column": 61
} | {
"line": 766,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : NumberField K\nF₁ F₂ : IntermediateField ℚ K\nn₁ n₂ : ℕ\ninst✝³ : NeZero n₁\ninst✝² : NeZero n₂\ninst✝¹ : IsCyclotomicExtension {n₁} ℚ ↥F₁\ninst✝ : IsCyclotomicExtension {n₂} ℚ ↥F₂\nζ₁ : ↥F₁\nhζ₁✝ : IsPrimitiveRoot ζ₁ n₁\nh₁ : ℤ[hζ₁✝.toInteger... | [] | exact isCoprime_differentIdeal_of_isCoprime_discr _ h_cpr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 58
} | {
"line": 124,
"column": 2
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nh : (logEmbedding K) (Additive.ofMul x) = 0\nw : InfinitePlace K\nhw : ¬w = w₀\n⊢ w ((algebraMap (𝓞 K) K) ↑x) = 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NumberField.InfinitePlace.instFun... | [] | exact mult_log_place_eq_zero.mp (congrFun h ⟨w, hw⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 58
} | {
"line": 124,
"column": 2
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nh : (logEmbedding K) (Additive.ofMul x) = 0\nw : InfinitePlace K\nhw : ¬w = w₀\n⊢ w ((algebraMap (𝓞 K) K) ↑x) = 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NumberField.InfinitePlace.instFun... | [] | exact mult_log_place_eq_zero.mp (congrFun h ⟨w, hw⟩) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 58
} | {
"line": 124,
"column": 2
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nh : (logEmbedding K) (Additive.ofMul x) = 0\nw : InfinitePlace K\nhw : ¬w = w₀\n⊢ w ((algebraMap (𝓞 K) K) ↑x) = 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NumberField.InfinitePlace.instFun... | [] | exact mult_log_place_eq_zero.mp (congrFun h ⟨w, hw⟩) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Fintype | {
"line": 28,
"column": 4
} | {
"line": 30,
"column": 8
} | {
"line": 32,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : Fintype R\ninst✝ : DecidableEq R\nh : Fintype.card R ≤ 2\nh✝ : Nontrivial R\n⊢ univ = {0, 1}",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"NeZero.one",
"Finset.univ",
"congrArg",
... | [] | refine (eq_of_subset_of_card_le (subset_univ _) ?_).symm
convert! h
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Fintype | {
"line": 28,
"column": 4
} | {
"line": 30,
"column": 8
} | {
"line": 32,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : Fintype R\ninst✝ : DecidableEq R\nh : Fintype.card R ≤ 2\nh✝ : Nontrivial R\n⊢ univ = {0, 1}",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"NeZero.one",
"Finset.univ",
"congrArg",
... | [] | refine (eq_of_subset_of_card_le (subset_univ _) ?_).symm
convert! h
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 732,
"column": 78
} | {
"line": 732,
"column": 83
} | {
"line": 732,
"column": 83
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTow... | [
"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTower A B L\nin... | ← hz' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 339,
"column": 6
} | {
"line": 340,
"column": 25
} | {
"line": 341,
"column": 4
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }\nv : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)\nw : { w // w ≠ w₀ }\n⊢ 0 ≤ |v w w| - v w w",
"ppTerm": "?... | [] | rw [sub_nonneg]
exact le_abs_self _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 339,
"column": 6
} | {
"line": 340,
"column": 25
} | {
"line": 341,
"column": 4
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }\nv : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)\nw : { w // w ≠ w₀ }\n⊢ 0 ≤ |v w w| - v w w",
"ppTerm": "?... | [] | rw [sub_nonneg]
exact le_abs_self _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 95
} | {
"line": 413,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na₁✝ a₂✝ : Additive ((𝓞 K)ˣ ⧸ torsion K)\nh :\n (QuotientGroup.kerLift (AddMonoidHom.toMultiplicativeRight (logEmbedding K)))\n ((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₁✝)) =\n (QuotientGroup.kerLift (AddMonoidHom.toMultip... | [] | exact (EmbeddingLike.apply_eq_iff_eq _).mp <| (QuotientGroup.kerLift_injective _).eq_iff.mp h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 486,
"column": 41
} | {
"line": 486,
"column": 57
} | {
"line": 486,
"column": 58
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ni : Fin (rank K)\n⊢ (logEmbedding K) (Additive.ofMul (fundSystem K i)) = ↑((logEmbeddingEquiv K) ((basisModTorsion K) i))",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Real",
"... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ni : Fin (rank K)\n⊢ (logEmbedding K) (Additive.ofMul (fundSystem K i)) = ↑((logEmbeddingEquiv K) (Additive.ofMul ↑(fundSystem K i)))"
] | ← fundSystem_mk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 499,
"column": 17
} | {
"line": 499,
"column": 38
} | {
"line": 499,
"column": 39
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx ζ : (𝓞 K)ˣ\nf : Fin (rank K) → ℤ\nhζ : ζ ∈ torsion K\nh : x = ζ * ∏ i, fundSystem K i ^ f i\n⊢ Additive.ofMul ↑(ζ * ∏ i, fundSystem K i ^ f i) = ∑ i, f i • Additive.ofMul ↑(fundSystem K i)",
"ppTerm": "?m.125",
"assigned": true,
"use... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx ζ : (𝓞 K)ˣ\nf : Fin (rank K) → ℤ\nhζ : ζ ∈ torsion K\nh : x = ζ * ∏ i, fundSystem K i ^ f i\n⊢ Additive.ofMul (↑ζ * ↑(∏ i, fundSystem K i ^ f i)) = ∑ i, f i • Additive.ofMul ↑(fundSystem K i)"
] | QuotientGroup.mk_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 510,
"column": 36
} | {
"line": 510,
"column": 57
} | {
"line": 510,
"column": 58
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nζ : (𝓞 K)ˣ := x * (∏ i, fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹\n⊢ ↑ζ = 1",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Semiring.... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nζ : (𝓞 K)ˣ := x * (∏ i, fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹\n⊢ ↑x * ↑(∏ i, fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹ = 1"
] | QuotientGroup.mk_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 517,
"column": 4
} | {
"line": 520,
"column": 36
} | {
"line": 522,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nζ : (𝓞 K)ˣ := x * (∏ i, fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹\nh_tors : ζ ∈ torsion K\n⊢ ∀ (y : ↥(torsion K) × (Fin (rank K) → ℤ)),\n (fun ζe ↦ x = ↑ζe.1 * ∏ i, fundSystem K i ^ ζe.2 i)... | [] | rintro ⟨⟨ζ', h_tors'⟩, η⟩ hf
simp only [ζ, ← fun_eq_repr K h_tors' hf, Prod.mk.injEq, Subtype.mk.injEq, and_true]
nth_rewrite 1 [hf]
rw [_root_.mul_inv_cancel_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 517,
"column": 4
} | {
"line": 520,
"column": 36
} | {
"line": 522,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nζ : (𝓞 K)ˣ := x * (∏ i, fundSystem K i ^ ((basisModTorsion K).repr (Additive.ofMul ↑x)) i)⁻¹\nh_tors : ζ ∈ torsion K\n⊢ ∀ (y : ↥(torsion K) × (Fin (rank K) → ℤ)),\n (fun ζe ↦ x = ↑ζe.1 * ∏ i, fundSystem K i ^ ζe.2 i)... | [] | rintro ⟨⟨ζ', h_tors'⟩, η⟩ hf
simp only [ζ, ← fun_eq_repr K h_tors' hf, Prod.mk.injEq, Subtype.mk.injEq, and_true]
nth_rewrite 1 [hf]
rw [_root_.mul_inv_cancel_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FLT.Three | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 69
} | {
"line": 114,
"column": 4
} | [
{
"pp": "a b c : ℤ\nha : a ≠ 0\nh3a : 3 ∣ a\nHgcd : {a, b, c}.gcd id = 1\nH : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3\nHF : a ^ 3 + b ^ 3 + c ^ 3 = 0\nx : ℤ\nhx : x = a ∨ x = b ∨ x = c\n⊢ 3 ∣ b",
"ppTerm": "?m.187",
"assigned": true,
"usedConstants": [
... | [
"a b c : ℤ\nha : a ≠ 0\nh3a : 3 ∣ a\nHgcd : {a, b, c}.gcd id = 1\nH : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3\nHF : a ^ 3 + b ^ 3 + c ^ 3 = 0\nx : ℤ\nhx : x = a ∨ x = b ∨ x = c\n⊢ {a, b, c}.gcd id = 1"
] | refine three_dvd_b_of_dvd_a_of_gcd_eq_one_of_case2 ha ?_ h3a HF H | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic | {
"line": 202,
"column": 14
} | {
"line": 202,
"column": 16
} | {
"line": 202,
"column": 16
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\na : F\nha : a = 0\n⊢ (quadraticChar F) a = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"CommSemiring.toCommMonoidWith... | [
"case pos\nF : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\na : F\nha : a = 0\n⊢ (quadraticChar F) 0 = 0"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 858,
"column": 23
} | {
"line": 858,
"column": 25
} | {
"line": 858,
"column": 25
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal ... | [
"A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal A\ninst✝² : ... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.NatInt | {
"line": 59,
"column": 2
} | {
"line": 60,
"column": 54
} | {
"line": 61,
"column": 2
} | [
{
"pp": "P : Ideal ℕ\n⊢ P.IsPrime ↔ P = ⊥ ∨ P = maximalIdeal ℕ ∨ ∃ p, Nat.Prime p ∧ P = span {p}",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Submodule",
"Nat.Prime",
"Semiring.toModule",
"Classical.or_iff_not_imp_right",
"CommSemiring.toS... | [
"case refine_1\nP : Ideal ℕ\n⊢ (P = ⊥ ∨ P = maximalIdeal ℕ ∨ ∃ p, Nat.Prime p ∧ P = span {p}) → P.IsPrime",
"case refine_2\nP : Ideal ℕ\nh : P.IsPrime\nh0 : ¬P = ⊥\nhsp : ¬∃ p, Nat.Prime p ∧ P = span {p}\nn : ℕ\nhn : n ∈ maximalIdeal ℕ\n⊢ n ∈ P"
] | refine .symm ⟨?_, fun h ↦ or_iff_not_imp_left.mpr fun h0 ↦ or_iff_not_imp_right.mpr fun hsp ↦
(le_maximalIdeal h.ne_top).antisymm fun n hn ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 889,
"column": 78
} | {
"line": 889,
"column": 83
} | {
"line": 889,
"column": 83
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal ... | [
"A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal A\ninst✝² : ... | ← hz' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 172,
"column": 12
} | {
"line": 172,
"column": 14
} | {
"line": 173,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\nH : (Multiset.map (fun v ↦ max (v x) 1) archAbsVal).prod * ∏ᶠ (v : ↑nonarchAbsVal), max (↑v x) 1 ≠ 0\na : ℝ\n⊢ a ∈ Multiset.map (fun v ↦ max (v x) 1) archAbsVal → a ≠ 0",
"ppTerm": "?m.89",
"assigned": true,
"usedConstant... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\nH : (Multiset.map (fun v ↦ max (v x) 1) archAbsVal).prod * ∏ᶠ (v : ↑nonarchAbsVal), max (↑v x) 1 ≠ 0\na : ℝ\nha : a ∈ Multiset.map (fun v ↦ max (v x) 1) archAbsVal\n⊢ a ≠ 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.Height.Basic | {
"line": 174,
"column": 8
} | {
"line": 174,
"column": 10
} | {
"line": 174,
"column": 10
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\nH : (Multiset.map (fun v ↦ max (v x) 1) archAbsVal).prod * ∏ᶠ (v : ↑nonarchAbsVal), max (↑v x) 1 ≠ 0\na : ℝ\nha : a = 0\n⊢ a ∉ Multiset.map (fun v ↦ max (v x) 1) archAbsVal",
"ppTerm": "?m.95",
"assigned": true,
"usedCons... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\nH : (Multiset.map (fun v ↦ max (v x) 1) archAbsVal).prod * ∏ᶠ (v : ↑nonarchAbsVal), max (↑v x) 1 ≠ 0\na : ℝ\nha : a = 0\n⊢ 0 ∉ Multiset.map (fun v ↦ max (v x) 1) archAbsVal"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 355,
"column": 4
} | {
"line": 355,
"column": 49
} | {
"line": 356,
"column": 4
} | [
{
"pp": "case inr.refine_1\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\nhx : x ≠ 0\ni : ι\nhi : x i ≠ 0\nhx' : (x i)⁻¹ • x ≠ 0\nv : AbsoluteValue K ℝ\nx✝ : v ∈ archAbsVal\n⊢ 1 ≤ ⨆ i_1, v (((x i)⁻¹ • x) i_1)",
"ppTerm": "?inr.refine_1",
"assi... | [
"case inr.refine_1\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\nhx : x ≠ 0\ni : ι\nhi : x i ≠ 0\nhx' : (x i)⁻¹ • x ≠ 0\nv : AbsoluteValue K ℝ\nx✝ : v ∈ archAbsVal\n⊢ 1 = v (((x i)⁻¹ • x) i)"
] | refine Finite.le_ciSup_of_le i <| le_of_eq ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 31,
"column": 2
} | {
"line": 31,
"column": 29
} | {
"line": 33,
"column": 0
} | [
{
"pp": "f : Fin 2 → ℝ\n⊢ iSup ![f 0, f 1] = max (![f 0, f 1] 0) (![f 0, f 1] 1)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Real",
"Lattice.toSemilatticeSup",
"iSup",
"SemilatticeSup.toMax",
"Fin.instOfNat",
"instOfNatNat",
"ConditionallyComp... | [] | exact (max_eq_iSup ..).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.FLT.Three | {
"line": 756,
"column": 19
} | {
"line": 756,
"column": 21
} | {
"line": 756,
"column": 22
} | [
{
"pp": "K : Type := CyclotomicField 3 ℚ\nhζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) 3 := IsCyclotomicExtension.zeta_spec 3 ℚ K\nthis : NumberField K\na b c : NumberField.RingOfIntegers K\nu : (NumberField.RingOfIntegers K)ˣ\nhc : c ≠ 0\n⊢ ¬hζ.toInteger - 1 ∣ a → ¬hζ.toInteger - 1 ∣ b → hζ.toIntege... | [
"K : Type := CyclotomicField 3 ℚ\nhζ : IsPrimitiveRoot (IsCyclotomicExtension.zeta 3 ℚ K) 3 := IsCyclotomicExtension.zeta_spec 3 ℚ K\nthis : NumberField K\na b c : NumberField.RingOfIntegers K\nu : (NumberField.RingOfIntegers K)ˣ\nhc : c ≠ 0\nha : ¬hζ.toInteger - 1 ∣ a\n⊢ ¬hζ.toInteger - 1 ∣ b → hζ.toInteger - 1 ∣ ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 223,
"column": 74
} | {
"line": 252,
"column": 30
} | {
"line": 254,
"column": 0
} | [
{
"pp": "s : ℝ\nhs : 1 < s\n⊢ termTSum s = 1 / (s - 1) - 1 / s * ∑' (n : ℕ), 1 / (↑n + 1) ^ s",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Filter.tendsto_atTop_add_const_right",
"Iff.mpr",
"zero_le",
"Filter.Tendsto.div_atTop",
"NonUnitalNonAssocCommRing.t... | [] | by
apply HasSum.tsum_eq
rw [hasSum_iff_tendsto_nat_of_nonneg (fun n ↦ term_nonneg (n + 1) s)]
change Tendsto (fun N ↦ termSum s N) atTop _
simp_rw [termSum_of_lt _ hs]
apply Tendsto.sub
· rw [show 𝓝 (1 / (s - 1)) = 𝓝 (1 / (s - 1) - 1 / (s - 1) * 0) by simp]
simp_rw [mul_sub, mul_one]
refine tendst... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Height.Basic | {
"line": 899,
"column": 65
} | {
"line": 900,
"column": 53
} | {
"line": 902,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\n⊢ logHeight₁ (-x) = logHeight₁ x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"Real",
"congrArg",
"Height.logHeight₁",
"Field.toDivisionRing",
"Divisi... | [] | by
simp [logHeight₁_eq_log_mulHeight₁, mulHeight₁_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RootsOfUnity.Lemmas | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 68
} | {
"line": 56,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nk n : ℕ\nμ : R\nn' : ℕ\nhn : k < n' + 1\nhμ : IsPrimitiveRoot μ (n' + 1)\n⊢ ∃ z ∈ ℤ[μ], ↑(n' + 1) = z * (μ - 1) ^ k",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"HMul.hMul",
"AddG... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nk n : ℕ\nμ : R\nm : ℕ\nhn : k < m + k + 1\nhμ : IsPrimitiveRoot μ (m + k + 1)\n⊢ ∃ z ∈ ℤ[μ], ↑(m + k + 1) = z * (μ - 1) ^ k"
] | obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le' (Nat.le_of_lt_succ hn) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 353,
"column": 50
} | {
"line": 353,
"column": 52
} | {
"line": 354,
"column": 6
} | [
{
"pp": "f : ℂ → ℂ := fun s ↦ riemannZeta s - 1 / (s - 1)\nC : ℂ\nhC : Tendsto f (𝓝[≠] 1) (𝓝 C)\na : ℝ\n⊢ a ∈ Ioi 1 → ↑a ∈ {1}ᶜ",
"ppTerm": "?m.151",
"assigned": true,
"usedConstants": [
"Real",
"Set.Ioi",
"Membership.mem",
"Real.instOne",
"One.toOfNat1",
"OfNat... | [
"f : ℂ → ℂ := fun s ↦ riemannZeta s - 1 / (s - 1)\nC : ℂ\nhC : Tendsto f (𝓝[≠] 1) (𝓝 C)\na : ℝ\nha : a ∈ Ioi 1\n⊢ ↑a ∈ {1}ᶜ"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 398,
"column": 31
} | {
"line": 398,
"column": 33
} | {
"line": 398,
"column": 34
} | [
{
"pp": "h : Tendsto (fun b ↦ (b.Gammaℝ - 1) / (b - 1)) (𝓝[≠] 1) (𝓝 (-(↑γ + Complex.log (4 * ↑π)) / 2))\nthis✝ : Tendsto ((fun b ↦ (b.Gammaℝ - 1) / (b - 1)) / Gammaℝ) (𝓝[≠] 1) (𝓝 (-(↑γ + Complex.log (4 * ↑π)) / 2))\nthis : {z | 0 < z.re} ∈ 𝓝 1\na : ℂ\n⊢ 0 < a.re → a ∈ {1}ᶜ → ((fun b ↦ (b.Gammaℝ - 1) / (b -... | [
"h : Tendsto (fun b ↦ (b.Gammaℝ - 1) / (b - 1)) (𝓝[≠] 1) (𝓝 (-(↑γ + Complex.log (4 * ↑π)) / 2))\nthis✝ : Tendsto ((fun b ↦ (b.Gammaℝ - 1) / (b - 1)) / Gammaℝ) (𝓝[≠] 1) (𝓝 (-(↑γ + Complex.log (4 * ↑π)) / 2))\nthis : {z | 0 < z.re} ∈ 𝓝 1\na : ℂ\nha : 0 < a.re\n⊢ a ∈ {1}ᶜ → ((fun b ↦ (b.Gammaℝ - 1) / (b - 1)) / G... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 168,
"column": 2
} | {
"line": 183,
"column": 41
} | {
"line": 185,
"column": 0
} | [
{
"pp": "F : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : Fintype F\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nχ φ : MulChar F R\nh : χ * φ ≠ 1\nψ : AddChar F R\n⊢ gaussSum (χ * φ) ψ * jacobiSum χ φ = gaussSum χ ψ * gaussSum φ ψ",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"C... | [] | rw [gaussSum_mul _ _ ψ, sum_eq_sum_sdiff_singleton_add (mem_univ (0 : F))]
conv =>
enter [2, 2, 2, x]
rw [zero_sub, neg_eq_neg_one_mul x, map_mul, mul_left_comm (χ x) (φ (-1)),
← MulChar.mul_apply, ψ.map_zero_eq_one, mul_one]
rw [← mul_sum _ _ (φ (-1)), MulChar.sum_eq_zero_of_ne_one h, mul_zero, add_z... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 168,
"column": 2
} | {
"line": 183,
"column": 41
} | {
"line": 185,
"column": 0
} | [
{
"pp": "F : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : Fintype F\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nχ φ : MulChar F R\nh : χ * φ ≠ 1\nψ : AddChar F R\n⊢ gaussSum (χ * φ) ψ * jacobiSum χ φ = gaussSum χ ψ * gaussSum φ ψ",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"C... | [] | rw [gaussSum_mul _ _ ψ, sum_eq_sum_sdiff_singleton_add (mem_univ (0 : F))]
conv =>
enter [2, 2, 2, x]
rw [zero_sub, neg_eq_neg_one_mul x, map_mul, mul_left_comm (χ x) (φ (-1)),
← MulChar.mul_apply, ψ.map_zero_eq_one, mul_one]
rw [← mul_sum _ _ (φ (-1)), MulChar.sum_eq_zero_of_ne_one h, mul_zero, add_z... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.NumberField | {
"line": 324,
"column": 97
} | {
"line": 327,
"column": 85
} | {
"line": 329,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : DecidableEq (InfinitePlace K)\nn : ℕ\nhn : n ≠ 0\nv : InfinitePlace K\n⊢ ↑n ^ (totalWeight K - 1) = (∏ w ∈ univ.erase v, ↑n ^ w.mult) * ↑n ^ (v.mult - 1)",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Finset.m... | [] | by
refine mul_right_cancel₀ (b := (n : ℝ)) (mod_cast hn) ?_
rw [pow_sub_one_mul (totalWeight_pos K).ne', totalWeight_eq_sum_mult, ← prod_pow_eq_pow_sum,
← prod_erase_mul _ _ (mem_univ v), ← pow_sub_one_mul v.mult_ne_zero, ← mul_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 63
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case e_a\nk : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\nthis : ↑(2 * k + 1)! = (2 * ↑k + 1) * Complex.Gamma (2 * ↑k + 1)\n⊢ (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / (2 * (2 * ↑k + 1) * Complex.Gamma (2 * ↑k + 1)) =\n (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * ↑k + 1) / ((2 * ↑k + 1) * 2 * Complex.Gamma (2 * ↑k... | [
"case e_a\nk : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\nthis : ↑(2 * k + 1)! = (2 * ↑k + 1) * Complex.Gamma (2 * ↑k + 1)\n⊢ (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / (2 * ↑(2 * k + 1) * Complex.Gamma ↑(2 * k + 1)) =\n (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k + 1) / (↑(2 * k + 1) * 2 * Complex.Gamma ↑(2 * k + 1))"
] | rw [(by simp : 2 * (k : ℂ) + 1 = ↑(2 * k + 1)), cpow_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 66
} | {
"line": 163,
"column": 2
} | [
{
"pp": "hp₀ : ∀ (p : Nat.Primes), 0 < (↑↑p)⁻¹\nhp₁ : ∀ (p : Nat.Primes), (↑↑p)⁻¹ < 1\nthis : Summable fun pk ↦ (↑↑pk.1)⁻¹ ^ (↑pk.2 + 3 / 2)\npk : Nat.Primes × ℕ\n⊢ 0 ≤ (if Nat.Prime (↑pk.1 ^ (pk.2 + 1 + 1)) then 0 else Λ (↑pk.1 ^ (pk.2 + 1 + 1))) / ↑↑pk.1 ^ (pk.2 + 1 + 1)",
"ppTerm": "?m.117",
"assigne... | [] | positivity [vonMangoldt_nonneg (n := (pk.1 : ℕ) ^ (pk.2 + 2))] | Mathlib.Tactic.Positivity._aux_Mathlib_Tactic_Positivity_Core___macroRules_Mathlib_Tactic_Positivity_positivity_1 | Mathlib.Tactic.Positivity.positivity |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 95,
"column": 6
} | {
"line": 106,
"column": 98
} | {
"line": 107,
"column": 4
} | [
{
"pp": "case inl\nf : ℕ → ℂ\nn : ℕ\nh : ∀ m ≤ n, f m = 0\nha : abscissaOfAbsConv f < ⊤\ny : ℝ\nhay : abscissaOfAbsConv f < ↑y\nhyt : ↑y < ⊤\nF : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x\nhF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0\nhF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 →... | [] | have H₀ : (0 : ℝ) ≤ k / (n + 1) := by positivity
have H₀' : (0 : ℝ) ≤ (n + 1) / k := by positivity
have H₁ : (k / (n + 1) : ℂ) = (k / (n + 1) : ℝ) := by push_cast; rfl
have H₂ : (n + 1) / k < (1 : ℝ) :=
(div_lt_one <| mod_cast n.succ_pos.trans H).mpr <| mod_cast H
simp only [Set.mem_ofPr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 95,
"column": 6
} | {
"line": 106,
"column": 98
} | {
"line": 107,
"column": 4
} | [
{
"pp": "case inl\nf : ℕ → ℂ\nn : ℕ\nh : ∀ m ≤ n, f m = 0\nha : abscissaOfAbsConv f < ⊤\ny : ℝ\nhay : abscissaOfAbsConv f < ↑y\nhyt : ↑y < ⊤\nF : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x\nhF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0\nhF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 →... | [] | have H₀ : (0 : ℝ) ≤ k / (n + 1) := by positivity
have H₀' : (0 : ℝ) ≤ (n + 1) / k := by positivity
have H₁ : (k / (n + 1) : ℂ) = (k / (n + 1) : ℝ) := by push_cast; rfl
have H₂ : (n + 1) / k < (1 : ℝ) :=
(div_lt_one <| mod_cast n.succ_pos.trans H).mpr <| mod_cast H
simp only [Set.mem_ofPr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ZetaValues | {
"line": 470,
"column": 4
} | {
"line": 471,
"column": 11
} | {
"line": 472,
"column": 2
} | [
{
"pp": "⊢ (fun n ↦ 1 / ↑n ^ (2 * 1 + 1) * Real.sin (2 * π * ↑n * (1 / 4))) = fun n ↦ 1 / ↑n ^ 3 * Real.sin (π * ↑n / 2)",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.RingNF.nnrat_rawCast",
"Eq.mpr",
"NonAs... | [] | ext1 n
ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ZetaValues | {
"line": 470,
"column": 4
} | {
"line": 471,
"column": 11
} | {
"line": 472,
"column": 2
} | [
{
"pp": "⊢ (fun n ↦ 1 / ↑n ^ (2 * 1 + 1) * Real.sin (2 * π * ↑n * (1 / 4))) = fun n ↦ 1 / ↑n ^ 3 * Real.sin (π * ↑n / 2)",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.RingNF.nnrat_rawCast",
"Eq.mpr",
"NonAs... | [] | ext1 n
ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 470,
"column": 6
} | {
"line": 470,
"column": 22
} | {
"line": 470,
"column": 22
} | [
{
"pp": "q a : ℕ\nhq : q ≠ 0\nh : a.Coprime q\n⊢ ∃ᶠ (p : ℕ) in atTop, Prime p ∧ p ≡ a [MOD q]",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Prime",
"congrArg",
"Filter.frequently_atTop",
"instArchimedeanNat",
"Preorder.toLE",
"inst... | [
"q a : ℕ\nhq : q ≠ 0\nh : a.Coprime q\n⊢ ∀ (a_1 : ℕ), ∃ b, a_1 ≤ b ∧ Prime b ∧ b ≡ a [MOD q]"
] | frequently_atTop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 469,
"column": 46
} | {
"line": 473,
"column": 26
} | {
"line": 475,
"column": 0
} | [
{
"pp": "q a : ℕ\nhq : q ≠ 0\nh : a.Coprime q\n⊢ ∃ᶠ (p : ℕ) in atTop, Prime p ∧ p ≡ a [MOD q]",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Prime",
"congrArg",
"Filter.frequently_atTop",
"instArchimedeanNat",
"Preorder.toLE",
"inst... | [] | by
rw [frequently_atTop]
intro n
obtain ⟨p, hn, hp, ha⟩ := forall_exists_prime_gt_and_modEq n hq h
exact ⟨p, hn.le, hp, ha⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 134,
"column": 2
} | {
"line": 176,
"column": 89
} | {
"line": 178,
"column": 0
} | [
{
"pp": "K : Type u_1\nΓ₀ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : Valued K Γ₀\ninst✝¹ : v.RankOne\ninst✝ : IsDiscreteValuationRing ↥𝒪[K]\n⊢ TotallyBounded Set.univ ↔ Finite 𝓀[K]",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Metric.vadd_... | [] | constructor
· intro H
obtain ⟨p, hp⟩ := IsDiscreteValuationRing.exists_irreducible 𝒪[K]
have := Metric.finite_approx_of_totallyBounded H ‖p‖ (norm_pos_iff.mpr hp.ne_zero)
simp only [Set.subset_univ, Set.univ_subset_iff, true_and] at this
obtain ⟨t, ht, ht'⟩ := this
rw [← Set.finite_univ_iff]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 134,
"column": 2
} | {
"line": 176,
"column": 89
} | {
"line": 178,
"column": 0
} | [
{
"pp": "K : Type u_1\nΓ₀ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : Valued K Γ₀\ninst✝¹ : v.RankOne\ninst✝ : IsDiscreteValuationRing ↥𝒪[K]\n⊢ TotallyBounded Set.univ ↔ Finite 𝓀[K]",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Metric.vadd_... | [] | constructor
· intro H
obtain ⟨p, hp⟩ := IsDiscreteValuationRing.exists_irreducible 𝒪[K]
have := Metric.finite_approx_of_totallyBounded H ‖p‖ (norm_pos_iff.mpr hp.ne_zero)
simp only [Set.subset_univ, Set.univ_subset_iff, true_and] at this
obtain ⟨t, ht, ht'⟩ := this
rw [← Set.finite_univ_iff]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 482,
"column": 52
} | {
"line": 482,
"column": 54
} | {
"line": 482,
"column": 54
} | [
{
"pp": "case inl\na : ℤ\nb : ℕ\nhb : Odd b\nha : a = ↑a.natAbs\n⊢ J(a | b) = J(a | b % (4 * a.natAbs))",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Nat.instMod",
"instHMod",
"instMulNat",
"instOfNa... | [
"case inl\na : ℤ\nb : ℕ\nhb : Odd b\nha : a = ↑a.natAbs\n⊢ J(a | b) = J(↑a.natAbs | b % (4 * a.natAbs))"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 482,
"column": 52
} | {
"line": 482,
"column": 54
} | {
"line": 482,
"column": 54
} | [
{
"pp": "case inr\na : ℤ\nb : ℕ\nhb : Odd b\nha : a = -↑a.natAbs\n⊢ J(a | b) = J(a | b % (4 * a.natAbs))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Nat.instMod",
"instHMod",
"Int.instNegInt",
"ins... | [
"case inr\na : ℤ\nb : ℕ\nhb : Odd b\nha : a = -↑a.natAbs\n⊢ J(a | b) = J(-↑a.natAbs | b % (4 * a.natAbs))"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 482,
"column": 70
} | {
"line": 482,
"column": 72
} | {
"line": 482,
"column": 72
} | [
{
"pp": "case inl\na : ℤ\nb : ℕ\nhb : Odd b\nha : a = ↑a.natAbs\n⊢ J(a | b) = J(↑a.natAbs | b % (4 * a.natAbs))",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Nat.instMod",
"instHMod",
"instMulNat",
"... | [
"case inl\na : ℤ\nb : ℕ\nhb : Odd b\nha : a = ↑a.natAbs\n⊢ J(↑a.natAbs | b) = J(↑a.natAbs | b % (4 * a.natAbs))"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 482,
"column": 70
} | {
"line": 482,
"column": 72
} | {
"line": 482,
"column": 72
} | [
{
"pp": "case inr\na : ℤ\nb : ℕ\nhb : Odd b\nha : a = -↑a.natAbs\n⊢ J(a | b) = J(-↑a.natAbs | b % (4 * a.natAbs))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Nat.instMod",
"instHMod",
"Int.instNegInt",
... | [
"case inr\na : ℤ\nb : ℕ\nhb : Odd b\nha : a = -↑a.natAbs\n⊢ J(-↑a.natAbs | b) = J(-↑a.natAbs | b % (4 * a.natAbs))"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 21
} | {
"line": 71,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\n⊢ IsCompact {x | (valuation K) x ≤ (valuation K) γ}",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrde... | [
"case pos\nK : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\nhγ : γ = 0\n⊢ IsCompact {x | (valuation K) x ≤ (valuation K) γ}",
"case neg\nK : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : I... | by_cases hγ : γ = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 310,
"column": 8
} | {
"line": 310,
"column": 31
} | {
"line": 310,
"column": 31
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Se... | [
"f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖... | Real.norm_of_nonneg hs' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MahlerMeasure | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 59
} | {
"line": 67,
"column": 4
} | [
{
"pp": "case refine_3\nn : ℕ\nB₁ B₂ : Fin (n + 1) → ℝ\n⊢ Function.LeftInverse (fun p ↦ ⟨(ofFn (n + 1)) ↑p, ⋯⟩) fun p ↦ ⟨(toFn (n + 1)) ↑p, ⋯⟩",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.MahlerMeasure.0.Polynomial.ncard_boxPoly._proof_1_7"
]... | [] | grind [boxPoly, ofFn_comp_toFn_eq_id_of_natDegree_lt] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.NumberTheory.MahlerMeasure | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 59
} | {
"line": 67,
"column": 4
} | [
{
"pp": "case refine_3\nn : ℕ\nB₁ B₂ : Fin (n + 1) → ℝ\n⊢ Function.LeftInverse (fun p ↦ ⟨(ofFn (n + 1)) ↑p, ⋯⟩) fun p ↦ ⟨(toFn (n + 1)) ↑p, ⋯⟩",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.MahlerMeasure.0.Polynomial.ncard_boxPoly._proof_1_7"
]... | [] | grind [boxPoly, ofFn_comp_toFn_eq_id_of_natDegree_lt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.MahlerMeasure | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 59
} | {
"line": 67,
"column": 4
} | [
{
"pp": "case refine_3\nn : ℕ\nB₁ B₂ : Fin (n + 1) → ℝ\n⊢ Function.LeftInverse (fun p ↦ ⟨(ofFn (n + 1)) ↑p, ⋯⟩) fun p ↦ ⟨(toFn (n + 1)) ↑p, ⋯⟩",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.MahlerMeasure.0.Polynomial.ncard_boxPoly._proof_1_7"
]... | [] | grind [boxPoly, ofFn_comp_toFn_eq_id_of_natDegree_lt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.BoundedAtCusp | {
"line": 32,
"column": 2
} | {
"line": 33,
"column": 60
} | {
"line": 35,
"column": 0
} | [
{
"pp": "g : GL (Fin 2) ℝ\nf : ℍ → ℂ\nk : ℤ\nhg : ↑g 1 0 = 0\nhf : Tendsto (fun x ↦ ‖f x‖) atImInfty (nhds 0)\n⊢ Tendsto (fun x ↦ ‖(f ∣[k] g) x‖) atImInfty (nhds 0)",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
"Norm.norm",
"Units.val",
... | [] | simpa [ModularForm.slash_def, denom, hg, mul_assoc]
using (hf.comp <| tendsto_smul_atImInfty hg).mul_const _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.ModularForms.Identities | {
"line": 63,
"column": 16
} | {
"line": 63,
"column": 39
} | {
"line": 63,
"column": 39
} | [
{
"pp": "case mpr.inv\nf : ℍ → ℂ\nΓ : Subgroup (GL (Fin 2) ℝ)\ns : Set (GL (Fin 2) ℝ)\nhΓ : Γ = Subgroup.closure s\nk : ℤ\nh : ∀ γ ∈ s, f ∣[k] γ = f\nγ : GL (Fin 2) ℝ\nhγ : γ ∈ Γ\nx : GL (Fin 2) ℝ\nhx : x ∈ Subgroup.closure s\nhf : f ∣[k] x = f\n⊢ (f ∣[k] x) ∣[k] x⁻¹ = f ∣[k] x",
"ppTerm": "?mpr.inv",
"... | [
"case mpr.inv\nf : ℍ → ℂ\nΓ : Subgroup (GL (Fin 2) ℝ)\ns : Set (GL (Fin 2) ℝ)\nhΓ : Γ = Subgroup.closure s\nk : ℤ\nh : ∀ γ ∈ s, f ∣[k] γ = f\nγ : GL (Fin 2) ℝ\nhγ : γ ∈ Γ\nx : GL (Fin 2) ℝ\nhx : x ∈ Subgroup.closure s\nhf : f ∣[k] x = f\n⊢ f ∣[k] (x * x⁻¹) = f ∣[k] x"
] | ← SlashAction.slash_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 86,
"column": 72
} | {
"line": 94,
"column": 69
} | {
"line": 96,
"column": 0
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nq : ℂ\nhq : ‖q‖ < 1\n⊢ DifferentiableAt ℂ (cuspFunction h f) q",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
... | [] | by
rcases eq_or_ne q 0 with rfl | hq'
· exact hfper.differentiableAt_cuspFunction_zero hh
(eventually_of_mem (preimage_mem_comap (Ioi_mem_atTop 0))
(fun z hz ↦ UpperHalfPlane.mdifferentiableAt_iff.mp (hfhol ⟨z, hz⟩)))
(hfbdd.comp_tendsto tendsto_comap_im_ofComplex)
· exact Periodic.qParam_righ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Modular | {
"line": 135,
"column": 2
} | {
"line": 152,
"column": 66
} | {
"line": 153,
"column": 2
} | [
{
"pp": "z : ℍ\nthis✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule\nπ₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0\nπ₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1\nf : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1\nf_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)\nthis : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1... | [
"z : ℍ\nthis✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule\nπ₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0\nπ₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1\nf : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1\nf_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)\nthis : (fun p ↦ normSq (↑(p 0) * ↑z + ↑(p 1))) = ⇑normS... | have hf : LinearMap.ker f = ⊥ := by
let g : ℂ →ₗ[ℝ] Fin 2 → ℝ :=
LinearMap.pi ![imLm, imLm.comp ((z : ℂ) • ((conjAe : ℂ →ₐ[ℝ] ℂ) : ℂ →ₗ[ℝ] ℂ))]
suffices ((z : ℂ).im⁻¹ • g).comp f = LinearMap.id by exact LinearMap.ker_eq_bot_of_inverse this
apply LinearMap.ext
intro c
have hz : (z : ℂ).im ≠ 0 :... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Modular | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 67
} | {
"line": 199,
"column": 2
} | [
{
"pp": "cd : Fin 2 → ℤ\nhcd : IsCoprime (cd 0) (cd 1)\nmB : ℝ → Matrix (Fin 2) (Fin 2) ℝ := fun t ↦ of ![![t, -↑1], Int.cast ∘ cd]\nhmB : Continuous mB\n⊢ Tendsto (fun g ↦ (lcRow0 cd) ↑((SpecialLinearGroup.map (Int.castRingHom ℝ)) ↑g)) cofinite (cocompact ℝ)",
"ppTerm": "?m.70",
"assigned": true,
"... | [
"cd : Fin 2 → ℤ\nhcd : IsCoprime (cd 0) (cd 1)\nmB : ℝ → Matrix (Fin 2) (Fin 2) ℝ := fun t ↦ of ![![t, -↑1], Int.cast ∘ cd]\nhmB : Continuous mB\n⊢ Tendsto (mB ∘ fun g ↦ (lcRow0 cd) ↑((SpecialLinearGroup.map (Int.castRingHom ℝ)) ↑g)) cofinite\n (cocompact (Matrix (Fin 2) (Fin 2) ℝ))"
] | refine Filter.Tendsto.of_tendsto_comp ?_ (comap_cocompact_le hmB) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 70
} | {
"line": 180,
"column": 0
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h f) 0\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\n⊢ (PowerSeries.coeff 0) (qExpansion h f) = valueAtInfty f",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"iteratedDeriv_zero",
"InnerProductSpace.toNorm... | [] | simp [qExpansion_coeff, cuspFunction_apply_zero hh hfanalytic hfper] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 70
} | {
"line": 180,
"column": 0
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h f) 0\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\n⊢ (PowerSeries.coeff 0) (qExpansion h f) = valueAtInfty f",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"iteratedDeriv_zero",
"InnerProductSpace.toNorm... | [] | simp [qExpansion_coeff, cuspFunction_apply_zero hh hfanalytic hfper] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 70
} | {
"line": 180,
"column": 0
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h f) 0\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\n⊢ (PowerSeries.coeff 0) (qExpansion h f) = valueAtInfty f",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"iteratedDeriv_zero",
"InnerProductSpace.toNorm... | [] | simp [qExpansion_coeff, cuspFunction_apply_zero hh hfanalytic hfper] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.Petersson | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 27
} | {
"line": 130,
"column": 2
} | [
{
"pp": "E : Type u_3\ninst✝ : NormedAddCommGroup E\nf : ℍ → E\nhf : ∃ c > 0, f =O[atImInfty] fun τ ↦ Real.exp (-c * τ.im)\n⊢ IsZeroAtImInfty f",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Real",
"HMul.hMul",
"Real.instZero",
"Asymptotics.IsBigO",
"UpperHa... | [
"E : Type u_3\ninst✝ : NormedAddCommGroup E\nf : ℍ → E\na : ℝ\nha : a > 0\nha' : f =O[atImInfty] fun τ ↦ Real.exp (-a * τ.im)\n⊢ IsZeroAtImInfty f"
] | obtain ⟨a, ha, ha'⟩ := hf | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 293,
"column": 53
} | {
"line": 293,
"column": 55
} | {
"line": 293,
"column": 56
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nc : ℕ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h f) 0\nhf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)\nH1 : HasFPowerSeriesOnBall (update (cuspFunction h f) 0 (c 0)) (FormalMultilinearSeries.ofScalars ℂ c) 0 1\nL1 : ContinuousAt (update (cuspFunction h f) 0 (... | [
"h : ℝ\nf : ℍ → ℂ\nc : ℕ → ℂ\nhh : 0 < h\nhfanalytic : AnalyticAt ℂ (cuspFunction h f) 0\nhf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)\nH1 : HasFPowerSeriesOnBall (update (cuspFunction h f) 0 (c 0)) (FormalMultilinearSeries.ofScalars ℂ c) 0 1\nL1 : ContinuousAt (update (cuspFunction h f) 0 (c 0)) 0\nL2 ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.ModularForms.LevelOne.Basic | {
"line": 76,
"column": 2
} | {
"line": 87,
"column": 69
} | {
"line": 89,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nk : ℤ\ninst✝ : ModularFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nhk : k ≤ 0\nf : F\n⊢ Set.EqOn (UpperHalfPlane.cuspFunction 1 ⇑f) (Function.const ℂ (UpperHalfPlane.cuspFunction 1 (⇑f) 0))\n (Metric.ball 0 1)",
"ppTerm": "?m.46",
"assigned":... | [] | refine eq_const_of_exists_le (fun q hq ↦ ?_) (exp_nonneg (-π)) ?_ (fun q hq ↦ ?_)
· exact (ModularFormClass.differentiableAt_cuspFunction f one_pos one_mem_strictPeriods_SL
(mem_ball_zero_iff.mp hq)).differentiableWithinAt
· simp [pi_pos]
· simp only [Metric.mem_closedBall, dist_zero_right]
rcases eq_or... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.LevelOne.Basic | {
"line": 76,
"column": 2
} | {
"line": 87,
"column": 69
} | {
"line": 89,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nk : ℤ\ninst✝ : ModularFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nhk : k ≤ 0\nf : F\n⊢ Set.EqOn (UpperHalfPlane.cuspFunction 1 ⇑f) (Function.const ℂ (UpperHalfPlane.cuspFunction 1 (⇑f) 0))\n (Metric.ball 0 1)",
"ppTerm": "?m.46",
"assigned":... | [] | refine eq_const_of_exists_le (fun q hq ↦ ?_) (exp_nonneg (-π)) ?_ (fun q hq ↦ ?_)
· exact (ModularFormClass.differentiableAt_cuspFunction f one_pos one_mem_strictPeriods_SL
(mem_ball_zero_iff.mp hq)).differentiableWithinAt
· simp [pi_pos]
· simp only [Metric.mem_closedBall, dist_zero_right]
rcases eq_or... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 42
} | {
"line": 101,
"column": 4
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhF𝒟 : ... | [
"case neg\nE : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhF𝒟 : ‖f τ‖ ≤ F * ... | refine le_trans ?_ <| le_max_right _ _ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 331,
"column": 2
} | {
"line": 336,
"column": 9
} | {
"line": 338,
"column": 2
} | [
{
"pp": "case e_f\nh : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nn : ℕ\nt : ℝ\nht : 0 < t\nR : ℝ := rexp (-2 * π * t / h)\nhR0 : 0 < R\nhR1 : R < 1\nu : ℝ\nτ : ℍ := { coe := ↑u + ↑t * ↑I, coe_im_pos := ⋯ }\n⊢ (2 * ↑π * Complex.I)⁻¹ * ↑(2 * π /... | [
"case e_f\nh : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nn : ℕ\nt : ℝ\nht : 0 < t\nR : ℝ := rexp (-2 * π * t / h)\nhR0 : 0 < R\nhR1 : R < 1\nu : ℝ\nτ : ℍ := { coe := ↑u + ↑t * ↑I, coe_im_pos := ⋯ }\nthis : circleMap 0 R (u * (2 * π / h)) = 𝕢 h ↑τ... | have : circleMap 0 R (u * (2 * π / h)) = 𝕢 h τ := by
simp only [circleMap, ofReal_exp, ← exp_add, zero_add, τ, R]
congr 1
push_cast
have := I_sq
grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 47
} | {
"line": 197,
"column": 2
} | [
{
"pp": "k : ℤ\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝⁴ : Γ.IsArithmetic\nF : Type u_2\nF' : Type u_3\nf : F\nf' : F'\ninst✝³ : FunLike F ℍ ℂ\ninst✝² : FunLike F' ℍ ℂ\ninst✝¹ : CuspFormClass F Γ k\ninst✝ : ModularFormClass F' Γ k\ng : SL(2, ℤ)\n⊢ Tendsto (fun τ ↦ petersson k (⇑f) (⇑f') (g • τ)) atImInfty (𝓝 0)",
... | [
"k : ℤ\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝⁴ : Γ.IsArithmetic\nF : Type u_2\nF' : Type u_3\nf : F\nf' : F'\ninst✝³ : FunLike F ℍ ℂ\ninst✝² : FunLike F' ℍ ℂ\ninst✝¹ : CuspFormClass F Γ k\ninst✝ : ModularFormClass F' Γ k\ng : SL(2, ℤ)\n⊢ Tendsto (fun τ ↦ petersson k (⇑f ∣[k] g) (⇑f' ∣[k] g) τ) atImInfty (𝓝 0)"
] | simp_rw [← UpperHalfPlane.petersson_slash_SL] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 140,
"column": 60
} | {
"line": 140,
"column": 82
} | {
"line": 141,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nr : 𝕜\nhr : ‖r‖ < 1\nh1 : ∀ (m : ℕ+), ‖r ^ ↑m‖ < 1\nh2 : ∀ (m : ℕ+), ∑' (n : ℕ+), r ^ (↑n * ↑m) = (1 - r ^ ↑m)⁻¹ - 1\n⊢ Tendsto (fun x ↦ (1 - r ^ ↑x)⁻¹ - 1) atTop (𝓝 ((1 - 0)⁻¹ - 1))",
"ppTerm": "?m.163",
"assigned": true,
"usedConstants"... | [
"𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nr : 𝕜\nhr : ‖r‖ < 1\nh1 : ∀ (m : ℕ+), ‖r ^ ↑m‖ < 1\nh2 : ∀ (m : ℕ+), ∑' (n : ℕ+), r ^ (↑n * ↑m) = (1 - r ^ ↑m)⁻¹ - 1\n⊢ Tendsto (fun x ↦ (1 - r ^ ↑x)⁻¹) atTop (𝓝 (1 - 0)⁻¹)"
] | tendsto_sub_const_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 66
} | {
"line": 164,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf : ℤ → α\na : α\n⊢ HasProd f a (symmetricIoc ℤ) ↔ Tendsto (fun N ↦ ∏ n ∈ Ioc (-↑N) ↑N, f n) atTop (𝓝 a)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"SummationFilter.mk",
"congrArg",
"Filter.ma... | [] | simp [HasProd, symmetricIoc, ← Nat.map_cast_int_atTop, comp_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 66
} | {
"line": 164,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf : ℤ → α\na : α\n⊢ HasProd f a (symmetricIoc ℤ) ↔ Tendsto (fun N ↦ ∏ n ∈ Ioc (-↑N) ↑N, f n) atTop (𝓝 a)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"SummationFilter.mk",
"congrArg",
"Filter.ma... | [] | simp [HasProd, symmetricIoc, ← Nat.map_cast_int_atTop, comp_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 66
} | {
"line": 164,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf : ℤ → α\na : α\n⊢ HasProd f a (symmetricIoc ℤ) ↔ Tendsto (fun N ↦ ∏ n ∈ Ioc (-↑N) ↑N, f n) atTop (𝓝 a)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"SummationFilter.mk",
"congrArg",
"Filter.ma... | [] | simp [HasProd, symmetricIoc, ← Nat.map_cast_int_atTop, comp_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 566,
"column": 76
} | {
"line": 566,
"column": 92
} | {
"line": 566,
"column": 92
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : T ^ ↑g 0 0 • S • z ∈ 𝒟\nhg'✝ : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 0\nhb : ↑g 0 1 = -1\nhz' : ‖↑z‖ = 1\nhg' : g = T⁻¹ * S\nhSre : (S • z).re = -z.re\nh : ↑g 0 0 = -1\nthis : |↑(-1) + -z.re| ≤ 1 / ... | [] | rw [hz', norm_ρ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Modular | {
"line": 566,
"column": 76
} | {
"line": 566,
"column": 92
} | {
"line": 566,
"column": 92
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : T ^ ↑g 0 0 • S • z ∈ 𝒟\nhg'✝ : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 0\nhb : ↑g 0 1 = -1\nhz' : ‖↑z‖ = 1\nhg' : g = T⁻¹ * S\nhSre : (S • z).re = -z.re\nh : ↑g 0 0 = -1\nthis : |↑(-1) + -z.re| ≤ 1 / ... | [] | rw [hz', norm_ρ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 566,
"column": 76
} | {
"line": 566,
"column": 92
} | {
"line": 566,
"column": 92
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : T ^ ↑g 0 0 • S • z ∈ 𝒟\nhg'✝ : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 0\nhb : ↑g 0 1 = -1\nhz' : ‖↑z‖ = 1\nhg' : g = T⁻¹ * S\nhSre : (S • z).re = -z.re\nh : ↑g 0 0 = -1\nthis : |↑(-1) + -z.re| ≤ 1 / ... | [] | rw [hz', norm_ρ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 68
} | {
"line": 69,
"column": 2
} | [
{
"pp": "q : ℂ\nhq : ‖q‖ < 1\n⊢ Multipliable fun n ↦ 1 - q ^ (n + 1)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"multipliable_one_add_of_summable",
"Complex.instNormedField",
"CommCStarAlgebra.toNormedCommRing",
"NormedDivisionRing.to_normOneClass",
"inst... | [
"q : ℂ\nhq : ‖q‖ < 1\n⊢ Summable fun i ↦ ‖-q ^ (i + 1)‖"
] | apply multipliable_one_add_of_summable (f := fun n ↦ -q ^ (n + 1)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 162,
"column": 8
} | {
"line": 162,
"column": 36
} | {
"line": 162,
"column": 37
} | [
{
"pp": "case hg\nz : ℍ\n⊢ HasSum (fun n ↦ ∑' (m : ℤ), (1 / (↑m * ↑z + ↑n) - 1 / (↑m * ↑z + ↑n + 1))) (-2 * ↑π * I / ↑z) (symmetricIco ℤ)",
"ppTerm": "?hg",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"Re... | [
"case hg\nz : ℍ\n⊢ Tendsto (fun N ↦ ∑ n ∈ Finset.Ico (-↑N) ↑N, ∑' (m : ℤ), (1 / (↑m * ↑z + ↑n) - 1 / (↑m * ↑z + ↑n + 1))) atTop\n (𝓝 (-2 * ↑π * I / ↑z))"
] | hasSum_symmetricIco_int_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 217,
"column": 12
} | {
"line": 217,
"column": 14
} | {
"line": 217,
"column": 14
} | [
{
"pp": "γ g : SL(2, ℤ)\nhx✝ : g ∈ Subgroup.closure {S, T}\nig : G2 ∣[2] g = G2 - D2 g\n⊢ (G2 ∣[2] g) ∣[2] g⁻¹ = (G2 - D2 g) ∣[2] g⁻¹",
"ppTerm": "?m.158",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.SpecialLinearGroup",
"congrArg",
"instDecidableEqFin",
"Eis... | [
"γ g : SL(2, ℤ)\nhx✝ : g ∈ Subgroup.closure {S, T}\nig : G2 ∣[2] g = G2 - D2 g\n⊢ (G2 - D2 g) ∣[2] g⁻¹ = (G2 - D2 g) ∣[2] g⁻¹"
] | ig | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.Derivative | {
"line": 214,
"column": 84
} | {
"line": 221,
"column": 41
} | {
"line": 223,
"column": 0
} | [
{
"pp": "k : ℤ\nF : ℍ → ℂ\nhF : MDiff F\nγ : SL(2, ℤ)\n⊢ D (F ∣[k] γ) =\n D F ∣[k + 2] γ - fun z ↦\n ↑k * (2 * ↑π * I)⁻¹ *\n (↑(↑γ 1 0) /\n denom (Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑z) *\n (F ∣[k] γ) z",
"ppTerm": "?m.7... | [] | by
have hdet : (γ : GL (Fin 2) ℝ).val.det = 1 := by
rw [← Matrix.GeneralLinearGroup.val_det_apply]; simp
ext z
have := congrFun
(normalizedDerivOfComplex_slash (k := k) hF (g := (γ : GL (Fin 2) ℝ)) (by grind)) z
rw [hdet] at this
simpa [ModularForm.SL_slash] using this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 204,
"column": 11
} | {
"line": 204,
"column": 39
} | {
"line": 204,
"column": 40
} | [
{
"pp": "z : ℍ\nm : ℤ\n⊢ HasSum (fun b ↦ 1 / (↑m * ↑z + ↑b) - 1 / (↑m * ↑z + ↑b + 1)) 0 (symmetricIco ℤ)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"HMul.hMul",
"UpperHalfPlane... | [
"z : ℍ\nm : ℤ\n⊢ Tendsto (fun N ↦ ∑ b ∈ Ico (-↑N) ↑N, (1 / (↑m * ↑z + ↑b) - 1 / (↑m * ↑z + ↑b + 1))) atTop (𝓝 0)"
] | hasSum_symmetricIco_int_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 278,
"column": 6
} | {
"line": 278,
"column": 78
} | {
"line": 279,
"column": 4
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\n⊢ (E hk) z =\n 1 +\n (riemannZeta ↑k)⁻¹ * (-2 * ↑π * I) ^ k / ↑(k - 1)! * ∑' (n : ℕ+), ↑((σ (k - 1)) ↑n) * cexp (2 * ↑π * I * ↑z) ^ ↑↑n",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"SlashInvariantForm",
"PNat.val",
... | [
"k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\n⊢ (1 / 2) • (eisensteinSeriesSIF 0 ↑k) z =\n 1 +\n (riemannZeta ↑k)⁻¹ * (-2 * ↑π * I) ^ k / ↑(k - 1)! * ∑' (n : ℕ+), ↑((σ (k - 1)) ↑n) * cexp (2 * ↑π * I * ↑z) ^ ↑↑n"
] | show E hk z = (1 / 2 : ℂ) • eisensteinSeriesSIF (N := 1) 0 k z from rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Modular | {
"line": 871,
"column": 61
} | {
"line": 871,
"column": 63
} | {
"line": 871,
"column": 64
} | [
{
"pp": "x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\na : ℝ\n⊢ 0 < a → a ∈ Set.Iio 1 → |(↑ofComplex (↑a * ↑x)).re| < 1 / 2",
"ppTerm": "?m.198",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.instZero",
"PartialOrder.toPreorder",
"SemilatticeInf... | [
"x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\na : ℝ\nha : 0 < a\n⊢ a ∈ Set.Iio 1 → |(↑ofComplex (↑a * ↑x)).re| < 1 / 2"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
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