module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.Fermat | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 7
} | {
"line": 125,
"column": 2
} | [
{
"pp": "a : ℕ\nha : 1 < a\nk m : ℕ\nhm : Odd m\nhn : 2 ^ k * m ≠ 0\nhP : Prime ((a ^ 2 ^ k) ^ m + 1)\n⊢ ∃ m_1, 2 ^ k * m = 2 ^ m_1",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Nat.instMonoid",
"instMulNat",
"instOfNatNat",
"NPow.toPow",
... | [
"case h\na : ℕ\nha : 1 < a\nk m : ℕ\nhm : Odd m\nhn : 2 ^ k * m ≠ 0\nhP : Prime ((a ^ 2 ^ k) ^ m + 1)\n⊢ 2 ^ k * m = 2 ^ k"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.FLT.Three | {
"line": 346,
"column": 4
} | {
"line": 346,
"column": 82
} | {
"line": 347,
"column": 4
} | [
{
"pp": "case inr.inl\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nh : λ ^ 2 ∣ S'.a + ↑η * S'.b\n⊢ ∃ a' b', a' ^ 3 + b' ^ 3 = ↑S'.u * S'.c ^ 3 ∧ IsCoprime a' b' ∧ ¬λ ∣ a' ∧ ¬λ ∣ b' ∧ λ ^ 2 ∣ a' + b'",
"ppT... | [
"case inr.inl.refine_1\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nh : λ ^ 2 ∣ S'.a + ↑η * S'.b\n⊢ S'.a ^ 3 + (↑η * S'.b) ^ 3 = ↑S'.u * S'.c ^ 3",
"case inr.inl.refine_2\nK : Type u_1\ninst✝² : Field K\nζ : K\n... | refine ⟨S'.a, η * S'.b, ?_, ?_, S'.ha, fun ⟨x, hx⟩ ↦ S'.hb ⟨η ^ 2 * x, ?_⟩, h⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.NatInt | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 88
} | {
"line": 102,
"column": 4
} | [
{
"pp": "case refine_1\ns✝ : LTSeries (PrimeSpectrum ℕ)\nhs : 2 < s✝.length\ns : RelSeries {(a, b) | a < b} := RelSeries.take s✝ ⟨3, ⋯⟩\nthis : NeZero s.length\nh1 : ⊥ < (s.toFun 1).asIdeal\n⊢ False",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"Fal... | [
"case refine_1.inl\ns✝ : LTSeries (PrimeSpectrum ℕ)\nhs : 2 < s✝.length\ns : RelSeries {(a, b) | a < b} := RelSeries.take s✝ ⟨3, ⋯⟩\nthis : NeZero s.length\nh1 : ⊥ < (s.toFun 1).asIdeal\nhmax : (s.toFun 1).asIdeal = maximalIdeal ℕ\n⊢ False",
"case refine_1.inr\ns✝ : LTSeries (PrimeSpectrum ℕ)\nhs : 2 < s✝.length\... | obtain hmax | ⟨p, hp, hsp⟩ := (Ideal.isPrime_nat_iff.mp (s 1).2).resolve_left h1.ne' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.Harmonic.Int | {
"line": 36,
"column": 8
} | {
"line": 36,
"column": 21
} | {
"line": 36,
"column": 21
} | [
{
"pp": "case succ.inr\nn : ℕ\nih : padicValRat 2 (harmonic n) = -↑(Nat.log 2 n)\nhn : n ≠ 0\n⊢ padicValRat 2 (harmonic (n + 1)) = -↑(Nat.log 2 (n + 1))",
"ppTerm": "?succ.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Rat",
"id",
"Int.instNegInt",
... | [
"case succ.inr\nn : ℕ\nih : padicValRat 2 (harmonic n) = -↑(Nat.log 2 n)\nhn : n ≠ 0\n⊢ padicValRat 2 (harmonic n + (↑(n + 1))⁻¹) = -↑(Nat.log 2 (n + 1))"
] | harmonic_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.Int | {
"line": 31,
"column": 2
} | {
"line": 43,
"column": 52
} | {
"line": 45,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ padicValRat 2 (harmonic n) = -↑(Nat.log 2 n)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"Rat.instOfNat",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.instIs... | [] | induction n with
| zero => simp
| succ n ih =>
rcases eq_or_ne n 0 with rfl | hn
· simp
rw [harmonic_succ]
have key : padicValRat 2 (harmonic n) ≠ padicValRat 2 (↑(n + 1))⁻¹ := by
rw [ih, padicValRat.inv, padicValRat.of_nat, Ne, neg_inj, Nat.cast_inj]
exact Nat.log_ne_padicValNat_succ hn... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.NumberTheory.Harmonic.Int | {
"line": 31,
"column": 2
} | {
"line": 43,
"column": 52
} | {
"line": 45,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ padicValRat 2 (harmonic n) = -↑(Nat.log 2 n)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"Rat.instOfNat",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.instIs... | [] | induction n with
| zero => simp
| succ n ih =>
rcases eq_or_ne n 0 with rfl | hn
· simp
rw [harmonic_succ]
have key : padicValRat 2 (harmonic n) ≠ padicValRat 2 (↑(n + 1))⁻¹ := by
rw [ih, padicValRat.inv, padicValRat.of_nat, Ne, neg_inj, Nat.cast_inj]
exact Nat.log_ne_padicValNat_succ hn... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Harmonic.Int | {
"line": 31,
"column": 2
} | {
"line": 43,
"column": 52
} | {
"line": 45,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ padicValRat 2 (harmonic n) = -↑(Nat.log 2 n)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"Rat.instOfNat",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.instIs... | [] | induction n with
| zero => simp
| succ n ih =>
rcases eq_or_ne n 0 with rfl | hn
· simp
rw [harmonic_succ]
have key : padicValRat 2 (harmonic n) ≠ padicValRat 2 (↑(n + 1))⁻¹ := by
rw [ih, padicValRat.inv, padicValRat.of_nat, Ne, neg_inj, Nat.cast_inj]
exact Nat.log_ne_padicValNat_succ hn... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FrobeniusNumber | {
"line": 133,
"column": 83
} | {
"line": 169,
"column": 43
} | {
"line": 171,
"column": 0
} | [
{
"pp": "s : Set ℕ\n⊢ ∃ t n, ↑t ⊆ s ∧ ∀ m ≥ n, setGcd s ∣ m → m ∈ span ↑t",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"_private.Mathlib.NumberTheory.FrobeniusNumber.0.Nat.exists_mem_span_nat_finset_of_ge._simp_1_6",
"Iff.mpr",
"_private.M... | [] | by
by_cases h0 : setGcd s = 0
· refine ⟨∅, 0, by simp, fun _ _ dvd ↦ by cases zero_dvd_iff.mp (h0 ▸ dvd); exact zero_mem _⟩
-- Write the gcd of `s` as a ℤ-linear combination of a finite subset `t`.
have ⟨t, hts, a, eq⟩ := (Submodule.mem_span_image_iff_exists_fun _).mp
(span_singleton_setGcd s ▸ mem_span_sin... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Height.Basic | {
"line": 423,
"column": 4
} | {
"line": 423,
"column": 26
} | {
"line": 424,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\ne : ↑(support x) ⊕ ↑(support x)ᶜ ≃ ι := Equiv.Set.sumCompl (support x)\ni : ↑(support x) ⊕ ↑(support x)ᶜ\n⊢ (x ∘ ⇑e) i = Sum.elim (fun i ↦ x ↑i) 0 i",
"ppTerm": "?m.51",
"assigned": true,
... | [
"K : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\ne : ↑(support x) ⊕ ↑(support x)ᶜ ≃ ι := Equiv.Set.sumCompl (support x)\ni : ↑(support x) ⊕ ↑(support x)ᶜ\n⊢ x (e i) = Sum.elim (fun i ↦ x ↑i) 0 i"
] | simp only [comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Height.Basic | {
"line": 611,
"column": 2
} | {
"line": 611,
"column": 58
} | {
"line": 613,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\n⊢ logHeight₁ x⁻¹ = logHeight₁ x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Real",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Height.l... | [] | simp only [logHeight₁_eq_log_mulHeight₁, mulHeight₁_inv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Height.Basic | {
"line": 611,
"column": 2
} | {
"line": 611,
"column": 58
} | {
"line": 613,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\n⊢ logHeight₁ x⁻¹ = logHeight₁ x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Real",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Height.l... | [] | simp only [logHeight₁_eq_log_mulHeight₁, mulHeight₁_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.Basic | {
"line": 611,
"column": 2
} | {
"line": 611,
"column": 58
} | {
"line": 613,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\n⊢ logHeight₁ x⁻¹ = logHeight₁ x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Real",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Height.l... | [] | simp only [logHeight₁_eq_log_mulHeight₁, mulHeight₁_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.Basic | {
"line": 710,
"column": 76
} | {
"line": 710,
"column": 100
} | {
"line": 711,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nx : ι → K\nhx : x ≠ 0\ny : ι' → K\nhy : y ≠ 0\nhxy : (fun a ↦ x a.1 * y a.2) ≠ 0\n⊢ (Multiset.map (fun v ↦ ⨆ i, v (x i.1 * y i.2)) archAbsVal).prod * ∏ᶠ (v : ↑nonarchAbsVal... | [
"K : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nx : ι → K\nhx : x ≠ 0\ny : ι' → K\nhy : y ≠ 0\nhxy : (fun a ↦ x a.1 * y a.2) ≠ 0\n⊢ (Multiset.map (fun v ↦ ⨆ i, v (x i.1 * y i.2)) archAbsVal).prod * ∏ᶠ (v : ↑nonarchAbsVal), ⨆ i, ↑v (... | ← Multiset.prod_map_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 183,
"column": 6
} | {
"line": 185,
"column": 18
} | {
"line": 186,
"column": 4
} | [
{
"pp": "case hf\nn : ℕ\nhn : 0 < n\ns : ℝ\nhs : 1 < s\nhv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x\n⊢ IntervalIntegrable (fun x ↦ x ^ (-s)) volume (↑n) (↑n + 1)",
"ppTerm": "?hf✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.m... | [] | · refine intervalIntegral.intervalIntegrable_rpow (Or.inr <| notMem_uIcc_of_lt ?_ ?_)
· exact_mod_cast hn
· linarith | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 183,
"column": 6
} | {
"line": 185,
"column": 18
} | {
"line": 186,
"column": 4
} | [
{
"pp": "case hg\nn : ℕ\nhn : 0 < n\ns : ℝ\nhs : 1 < s\nhv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x\n⊢ IntervalIntegrable (fun x ↦ x ^ (-(s + 1))) volume (↑n) (↑n + 1)",
"ppTerm": "?hg",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [] | · refine intervalIntegral.intervalIntegrable_rpow (Or.inr <| notMem_uIcc_of_lt ?_ ?_)
· exact_mod_cast hn
· linarith | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Height.NumberField | {
"line": 256,
"column": 2
} | {
"line": 259,
"column": 27
} | {
"line": 260,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nhx : IsAlgebraic ℤ x\nm : ℕ\nr : 𝓞 K\nhm : m ≠ 0\nhmr : m • x = ↑r\nn : ℕ := (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))\nhndef : n = (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAd... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nhx : IsAlgebraic ℤ x\nm : ℕ\nr : 𝓞 K\nhm : m ≠ 0\nhmr : m • x = ↑r\nn : ℕ := (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (span {↑m, r}))\nhndef : n = (Submodule.toAddSubgroup (span {↑m})).relIndex (Submodule.toAddSubgroup (s... | have ha : n * x = a := by
refine mul_left_cancel₀ (mod_cast hm : (m : K) ≠ 0) ?_
rw [mul_left_comm, ← nsmul_eq_mul m, hmr]
exact_mod_cast ha'.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 233,
"column": 54
} | {
"line": 233,
"column": 66
} | {
"line": 233,
"column": 67
} | [
{
"pp": "F : Type u_1\nF' : Type u_2\ninst✝² : Fintype F\ninst✝¹ : Field F\ninst✝ : Field F'\nh : ringChar F' ≠ ringChar F\nχ φ : MulChar F F'\nhχ : χ ≠ 1\nhφ : φ ≠ 1\nhχφ : χ * φ ≠ 1\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nψ : PrimitiveAddChar F F' := FiniteField.primitiveC... | [
"F : Type u_1\nF' : Type u_2\ninst✝² : Fintype F\ninst✝¹ : Field F\ninst✝ : Field F'\nh : ringChar F' ≠ ringChar F\nχ φ : MulChar F F'\nhχ : χ ≠ 1\nhφ : φ ≠ 1\nhχφ : χ * φ ≠ 1\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' h\n... | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 547,
"column": 48
} | {
"line": 547,
"column": 58
} | {
"line": 547,
"column": 59
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\nf : R[X]\ninst✝¹ : Algebra R S\nh : IsAdjoinRootMonic S f\ninst✝ : Nontrivial S\nx : R\ni : ℕ\n⊢ (x • h.coeff 1) i = Pi.single 0 x i",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.mod... | [
"R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\nf : R[X]\ninst✝¹ : Algebra R S\nh : IsAdjoinRootMonic S f\ninst✝ : Nontrivial S\nx : R\ni : ℕ\n⊢ (x • Pi.single 0 1) i = Pi.single 0 x i"
] | coeff_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 299,
"column": 26
} | {
"line": 299,
"column": 39
} | {
"line": 299,
"column": 40
} | [
{
"pp": "case pos\nF : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype F\nn : ℕ\nhn : 2 < n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nq : ℕ\nhq : Fintype.card F = n * q + 1\nz₁ : R\nhz₁ : z₁ ∈ ℤ[μ]\nHz₁ : ↑n = z₁ * (μ... | [
"case pos\nF : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype F\nn : ℕ\nhn : 2 < n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nq : ℕ\nhq : Fintype.card F = n * q + 1\nz₁ : R\nhz₁ : z₁ ∈ ℤ[μ]\nHz₁ : ↑n = z₁ * (μ - 1) ^ 2\nh... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 315,
"column": 42
} | {
"line": 315,
"column": 55
} | {
"line": 315,
"column": 56
} | [
{
"pp": "case neg.refine_2\nF : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype F\nn : ℕ\nhn : 2 < n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nq : ℕ\nhq : Fintype.card F = n * q + 1\nz₁ : R\nhz₁ : z₁ ∈ ℤ[μ]\nHz₁ : ↑n ... | [
"case neg.refine_2\nF : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype F\nn : ℕ\nhn : 2 < n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nq : ℕ\nhq : Fintype.card F = n * q + 1\nz₁ : R\nhz₁ : z₁ ∈ ℤ[μ]\nHz₁ : ↑n = z₁ * (μ - ... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 77
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case inl\nN : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\ns : ℂ\nhs : s ≠ 1 ∨ ∑ j, Φ j = 0\nhs' : s ≠ 1\n⊢ DifferentiableAt ℂ (fun s ↦ ∑ j, Φ j * hurwitzZeta (toAddCircle j) s) s",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"InnerProductSpace.... | [
"case inr\nN : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhs : 1 ≠ 1 ∨ ∑ j, Φ j = 0\n⊢ DifferentiableAt ℂ (fun s ↦ ∑ j, Φ j * hurwitzZeta (toAddCircle j) s) 1"
] | · exact .fun_sum fun j _ ↦ (differentiableAt_hurwitzZeta _ hs').const_mul _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 29
} | {
"line": 175,
"column": 2
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ZMod N\ns : ℂ\nhjs : j ≠ 0 ∨ s ≠ 1\nU : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ\n⊢ LFunction (fun k ↦ 𝕖 (j * k)) s = expZeta (toAddCircle j) s",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Real",
"setOf",
"Real.instLT",
... | [
"N : ℕ\ninst✝ : NeZero N\nj : ZMod N\ns : ℂ\nhjs : j ≠ 0 ∨ s ≠ 1\nU : Set ℂ := if j = 0 then {z | z ≠ 1} else Set.univ\nV : Set ℂ := {z | 1 < z.re}\n⊢ LFunction (fun k ↦ 𝕖 (j * k)) s = expZeta (toAddCircle j) s"
] | let V := {z : ℂ | 1 < re z} | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 24
} | {
"line": 216,
"column": 2
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\ns : ℂ\nhs : Φ 0 = 0 ∨ s ≠ 1\nthis : ∀ (j : ZMod N), Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s = Φ j * expZeta (toAddCircle (-j)) s\n⊢ ∑ i, ↑N ^ (-s) * (𝓕 Φ i * hurwitzZeta (toAddCircle i) s) =\n ∑ x, ∑ i, Φ x * (↑N ^ (-s) * (𝕖 (-x * i) * hurwitzZeta (toAddCir... | [
"N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\ns : ℂ\nhs : Φ 0 = 0 ∨ s ≠ 1\nthis : ∀ (j : ZMod N), Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s = Φ j * expZeta (toAddCircle (-j)) s\n⊢ ∑ i, ↑N ^ (-s) * ((fun k ↦ ∑ j, 𝕖 (-(j * k)) • Φ j) i * hurwitzZeta (toAddCircle i) s) =\n ∑ y, ∑ x, Φ x * (↑N ^ (-s) * (𝕖 (-x * y) * hur... | rw [dft_def, sum_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ZetaValues | {
"line": 236,
"column": 18
} | {
"line": 236,
"column": 53
} | {
"line": 237,
"column": 2
} | [
{
"pp": "case pos\nk : ℕ\nn : ℤ\nhn : n ≠ 0\nh✝ : k = 1\n⊢ ↑1 = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"eq_self",
"of_eq_true",
"One.toOfNat1",
"Complex",
"OfNat.ofNat",
"Eq",
"Complex.instOne"
],
"usedFVars": [],
"usedGoals"... | [] | simp only [ofReal_one, ofReal_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ZetaValues | {
"line": 236,
"column": 18
} | {
"line": 236,
"column": 53
} | {
"line": 237,
"column": 2
} | [
{
"pp": "case neg\nk : ℕ\nn : ℤ\nhn : n ≠ 0\nh✝ : ¬k = 1\n⊢ ↑0 = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Complex.instZero",
"eq_self",
"of_eq_true",
"Zero.toOfNat0",
"Complex",
"OfNat.ofNat",
"Eq"
],
"usedFVars": [],
"usedGoa... | [] | simp only [ofReal_one, ofReal_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ZetaValues | {
"line": 260,
"column": 36
} | {
"line": 260,
"column": 49
} | {
"line": 260,
"column": 50
} | [
{
"pp": "case inr.refine_2\nk✝ : ℕ\nhk✝ : k✝ ≠ 0\nn : ℤ\nhn : n ≠ 0\nk : ℕ\nhk : 1 ≤ k\nh'k : bernoulliFourierCoeff k n = -↑k ! / (2 * ↑π * I * ↑n) ^ k\n⊢ 1 / (-2 * ↑π * I * ↑n) * -(↑(k + 1) * (-↑k ! / (2 * ↑π * I * ↑n) ^ k)) =\n -↑((k + 1) * k !) / (2 * ↑π * I * ↑n) ^ (k + 1)",
"ppTerm": "?inr.refine_2"... | [
"case inr.refine_2\nk✝ : ℕ\nhk✝ : k✝ ≠ 0\nn : ℤ\nhn : n ≠ 0\nk : ℕ\nhk : 1 ≤ k\nh'k : bernoulliFourierCoeff k n = -↑k ! / (2 * ↑π * I * ↑n) ^ k\n⊢ 1 / (-2 * ↑π * I * ↑n) * -(↑(k + 1) * (-↑k ! / (2 * ↑π * I * ↑n) ^ k)) =\n -(↑(k + 1) * ↑k !) / (2 * ↑π * I * ↑n) ^ (k + 1)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 68,
"column": 4
} | {
"line": 69,
"column": 7
} | {
"line": 71,
"column": 0
} | [
{
"pp": "k : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\n⊢ 1 < (2 * ↑k).re",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"Nat.cast_mul._simp_1",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"HMul.hMul",
"FloorRin... | [] | norm_cast
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 68,
"column": 4
} | {
"line": 69,
"column": 7
} | {
"line": 71,
"column": 0
} | [
{
"pp": "k : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\n⊢ 1 < (2 * ↑k).re",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"Nat.cast_mul._simp_1",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"HMul.hMul",
"FloorRin... | [] | norm_cast
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 94,
"column": 4
} | {
"line": 95,
"column": 7
} | {
"line": 97,
"column": 0
} | [
{
"pp": "k : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\n⊢ 1 < (2 * ↑k + 1).re",
"ppTerm": "?m.131",
"assigned": true,
"usedConstants": [
"Nat.cast_mul._simp_1",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"HMul.hMul",
"Floo... | [] | norm_cast
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 94,
"column": 4
} | {
"line": 95,
"column": 7
} | {
"line": 97,
"column": 0
} | [
{
"pp": "k : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\n⊢ 1 < (2 * ↑k + 1).re",
"ppTerm": "?m.131",
"assigned": true,
"usedConstants": [
"Nat.cast_mul._simp_1",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"HMul.hMul",
"Floo... | [] | norm_cast
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 126,
"column": 91
} | {
"line": 142,
"column": 21
} | {
"line": 144,
"column": 0
} | [
{
"pp": "k : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\n⊢ hurwitzZetaEven (↑x) (1 - 2 * ↑k) =\n -1 / (2 * ↑k) * Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Nat.cas... | [] | by
have h1 (n : ℕ) : (2 * k : ℂ) ≠ -n := by
rw [← Int.cast_ofNat, ← Int.cast_natCast, ← Int.cast_mul, ← Int.cast_natCast n, ← Int.cast_neg,
Ne, Int.cast_inj, ← Ne]
refine ne_of_gt ((neg_nonpos_of_nonneg n.cast_nonneg).trans_lt (mul_pos two_pos ?_))
exact Nat.cast_pos.mpr (Nat.pos_of_ne_zero hk)
ha... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 69
} | {
"line": 205,
"column": 2
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nχ : DirichletCharacter ℂ N\nhχ : χ ^ 2 = 1\nχ_ne : χ ≠ 1\nhL : LFunction χ 1 = 0\n⊢ False",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.LSeries.Nonvanishing.0.DirichletCharacter.BadChar.mk",
"_private.Mathlib.Nu... | [
"N : ℕ\ninst✝ : NeZero N\nχ : DirichletCharacter ℂ N\nhχ : χ ^ 2 = 1\nχ_ne : χ ≠ 1\nhL : LFunction χ 1 = 0\nB : BadChar N := { χ := χ, χ_ne := χ_ne, χ_sq := hχ, hχ := hL }\n⊢ False"
] | let B : BadChar N := { χ := χ, χ_sq := hχ, hχ := hL, χ_ne := χ_ne } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.ZetaValues | {
"line": 473,
"column": 23
} | {
"line": 473,
"column": 43
} | {
"line": 473,
"column": 44
} | [
{
"pp": "this : 1 / 4 = (algebraMap ℚ ℝ) (1 / 4)\n⊢ (-1) ^ (1 + 1) * (2 ^ (2 * 1 + 1) * π ^ (2 * 1 + 1)) / 2 / ↑(2 * 1 + 1)! *\n Polynomial.eval ((algebraMap ℚ ℝ) (1 / 4)) (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * 1 + 1))) =\n π ^ 3 / 32",
"ppTerm": "?m.146",
"assigned": true,
... | [
"this : 1 / 4 = (algebraMap ℚ ℝ) (1 / 4)\n⊢ (-1) ^ (1 + 1) * (2 ^ (2 * 1 + 1) * π ^ (2 * 1 + 1)) / 2 / ↑(2 * 1 + 1)! *\n Polynomial.eval₂ (algebraMap ℚ ℝ) ((algebraMap ℚ ℝ) (1 / 4)) (Polynomial.bernoulli (2 * 1 + 1)) =\n π ^ 3 / 32"
] | Polynomial.eval_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 63,
"column": 4
} | {
"line": 64,
"column": 83
} | {
"line": 65,
"column": 4
} | [
{
"pp": "case refine_1\nf : ℕ → ℂ\nr : ℝ\ns : ℂ\nhf : f 0 = 0\nhO : (fun n ↦ ∑ k ∈ Icc 1 n, ‖f k‖) =O[atTop] fun n ↦ ↑n ^ r\nhr : 0 ≤ r\nhs : r < s.re\nh₁ : -s ≠ 0\nh₂ : (-s).re + r ≤ 0\nh₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ‖↑x ^ (-s)‖) t\nh₄ : (deriv fun t ↦ ‖↑t ^ (-s)‖) =ᶠ[atTop] fun t ↦ -s.re * ... | [
"case refine_1\nf : ℕ → ℂ\nr : ℝ\ns : ℂ\nhf : f 0 = 0\nhO : (fun n ↦ ∑ k ∈ Icc 1 n, ‖f k‖) =O[atTop] fun n ↦ ↑n ^ r\nhr : 0 ≤ r\nhs : r < s.re\nh₁ : -s ≠ 0\nh₂ : (-s).re + r ≤ 0\nh₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ‖↑x ^ (-s)‖) t\nh₄ : (deriv fun t ↦ ‖↑t ^ (-s)‖) =ᶠ[atTop] fun t ↦ -s.re * t ^ (-(s.re ... | refine (Iff.mpr integrableOn_Ici_iff_integrableOn_Ioi
(integrableOn_Ioi_deriv_norm_ofReal_cpow zero_lt_one ?_)).locallyIntegrableOn | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 339,
"column": 4
} | {
"line": 339,
"column": 84
} | {
"line": 341,
"column": 0
} | [
{
"pp": "case inr\nq : ℕ\na : ZMod q\ninst✝ : NeZero q\ns : ℂ\nhs : s ∈ {s | 1 ≤ s.re}\nhs₁ : s ≠ 1\n⊢ ∀ (χ : DirichletCharacter ℂ q), ¬LFunction χ s = 0",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"ZMod.commRing",
"MulChar.hasOne",
"... | [] | exact fun χ ↦ LFunction_ne_zero_of_one_le_re χ (.inr hs₁) <| Set.mem_setOf.mp hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 42
} | {
"line": 375,
"column": 2
} | [
{
"pp": "q : ℕ\na : ZMod q\ninst✝ : NeZero q\nha : IsUnit a\nx : ℝ\nhx : 1 < x\n⊢ LFunctionResidueClassAux a ↑x = ↑(LFunctionResidueClassAux a ↑x).re",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"HSub.hSub",
"Complex.instDi... | [
"q : ℕ\na : ZMod q\ninst✝ : NeZero q\nha : IsUnit a\nx : ℝ\nhx : 1 < x\n⊢ (fun s ↦ L (fun n ↦ ↑(residueClass a n)) s - (↑q.totient)⁻¹ / (s - 1)) ↑x =\n ↑((fun s ↦ L (fun n ↦ ↑(residueClass a n)) s - (↑q.totient)⁻¹ / (s - 1)) ↑x).re"
] | rw [eqOn_LFunctionResidueClassAux ha hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 377,
"column": 17
} | {
"line": 378,
"column": 72
} | {
"line": 378,
"column": 72
} | [
{
"pp": "case e_a\nq : ℕ\na : ZMod q\ninst✝ : NeZero q\nha : IsUnit a\nx : ℝ\nhx : 1 < x\n⊢ ∑' (n : ℕ), term (fun n ↦ ↑(residueClass a n)) (↑x) n = ↑(∑' (n : ℕ), term (fun n ↦ ↑(residueClass a n)) (↑x) n).re",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneERe... | [
"case e_a\nq : ℕ\na : ZMod q\ninst✝ : NeZero q\nha : IsUnit a\nx : ℝ\nhx : 1 < x\n⊢ ∑' (n : ℕ), term (fun n ↦ ↑(residueClass a n)) (↑x) n =\n ↑(∑' (a_1 : ℕ), (term (fun n ↦ ↑(residueClass a n)) (↑x) a_1).re)"
] | re_tsum <| LSeriesSummable_of_abscissaOfAbsConv_lt_re <|
(abscissaOfAbsConv_residueClass_le_one a).trans_lt <| by norm_cast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 225,
"column": 2
} | {
"line": 226,
"column": 73
} | {
"line": 228,
"column": 0
} | [
{
"pp": "f g : ℕ → ℂ\nhf : abscissaOfAbsConv f < ⊤\nhg : abscissaOfAbsConv g < ⊤\nh : (fun x ↦ LSeries f ↑x) =ᶠ[atTop] fun x ↦ LSeries g ↑x\nn : ℕ\nhn : n ≠ 0\nhsub : (fun x ↦ LSeries (f - g) ↑x) =ᶠ[atTop] 0\nha : abscissaOfAbsConv (f - g) ≠ ⊤\n⊢ f n = g n",
"ppTerm": "?m.56",
"assigned": true,
"use... | [] | simpa only [Pi.sub_apply, sub_eq_zero]
using (LSeries_eventually_eq_zero_iff'.mp hsub).resolve_right ha n hn | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 245,
"column": 12
} | {
"line": 245,
"column": 38
} | {
"line": 246,
"column": 2
} | [
{
"pp": "case zero\nf : ℕ → ℂ\nhf : f 0 = 0 ∧ abscissaOfAbsConv f < ⊤\ng : ℕ → ℂ\nhg : g 0 = 0 ∧ abscissaOfAbsConv g < ⊤\nh : ∀ (n : ℕ), n ≠ 0 → f n = g n\n⊢ f 0 = g 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"EReal",
... | [] | exact hf.1.trans hg.1.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 245,
"column": 12
} | {
"line": 245,
"column": 38
} | {
"line": 246,
"column": 2
} | [
{
"pp": "case zero\nf : ℕ → ℂ\nhf : f 0 = 0 ∧ abscissaOfAbsConv f < ⊤\ng : ℕ → ℂ\nhg : g 0 = 0 ∧ abscissaOfAbsConv g < ⊤\nh : ∀ (n : ℕ), n ≠ 0 → f n = g n\n⊢ f 0 = g 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"EReal",
... | [] | exact hf.1.trans hg.1.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 245,
"column": 12
} | {
"line": 245,
"column": 38
} | {
"line": 246,
"column": 2
} | [
{
"pp": "case zero\nf : ℕ → ℂ\nhf : f 0 = 0 ∧ abscissaOfAbsConv f < ⊤\ng : ℕ → ℂ\nhg : g 0 = 0 ∧ abscissaOfAbsConv g < ⊤\nh : ∀ (n : ℕ), n ≠ 0 → f n = g n\n⊢ f 0 = g 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"EReal",
... | [] | exact hf.1.trans hg.1.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 97
} | {
"line": 127,
"column": 4
} | [
{
"pp": "case convert_2\nf : ℕ → ℂ\nhf : f 0 = 0\nr : ℝ\nhr : 0 ≤ r\ns : ℂ\nhs : r < s.re\nhS : LSeriesSummable f s\nh₁ : (-s - 1).re + r < -1\nh₂ : s ≠ 0\nh₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ↑x ^ (-s)) t\nh₄ : ∀ (n : ℕ), ∑ k ∈ Icc 0 n, f k = ∑ k ∈ Icc 1 n, f k\nhO : (fun n ↦ ∑ k ∈ Icc 0 n, f k) =... | [
"case convert_2.refine_1\nf : ℕ → ℂ\nhf : f 0 = 0\nr : ℝ\nhr : 0 ≤ r\ns : ℂ\nhs : r < s.re\nhS : LSeriesSummable f s\nh₁ : (-s - 1).re + r < -1\nh₂ : s ≠ 0\nh₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ↑x ^ (-s)) t\nh₄ : ∀ (n : ℕ), ∑ k ∈ Icc 0 n, f k = ∑ k ∈ Icc 1 n, f k\nhO : (fun n ↦ ∑ k ∈ Icc 0 n, f k) =O[a... | refine (IsBigO.mul_atTop_rpow_natCast_of_isBigO_rpow (-s.re) _ _ ?_ hO ?_).trans_tendsto hlim | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 16
} | {
"line": 123,
"column": 17
} | [
{
"pp": "a : ℤ\nb₁ b₂ : ℕ\nhb₁ : b₁ ≠ 0\nhb₂ : b₂ ≠ 0\n⊢ J(a | b₁ * b₂) = J(a | b₁) * J(a | b₂)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"jacobiSym.eq_1",
"Nat.Prime",
"Nat.prime_of_mem_primeFactorsList",
"HMul.hMul",
"congrArg",
"... | [
"a : ℤ\nb₁ b₂ : ℕ\nhb₁ : b₁ ≠ 0\nhb₂ : b₂ ≠ 0\n⊢ (List.pmap (fun p pp ↦ legendreSym p a) (b₁ * b₂).primeFactorsList ⋯).prod = J(a | b₁) * J(a | b₂)"
] | jacobiSym, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 155,
"column": 11
} | {
"line": 155,
"column": 21
} | {
"line": 155,
"column": 22
} | [
{
"pp": "a₁ a₂ : ℤ\nb : ℕ\n⊢ J(a₁ * a₂ | b) = J(a₁ | b) * J(a₂ | b)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"id",
"Int",
"Int.instMul",
"Eq",
"jacobiSym",
"instHMul"
],
"usedFVars": [
"a₁",
"a₂",
"b"
... | [
"a₁ a₂ : ℤ\nb : ℕ\n⊢ (List.pmap (fun p pp ↦ legendreSym p (a₁ * a₂)) b.primeFactorsList ⋯).prod =\n (List.pmap (fun p pp ↦ legendreSym p a₁) b.primeFactorsList ⋯).prod *\n (List.pmap (fun p pp ↦ legendreSym p a₂) b.primeFactorsList ⋯).prod"
] | jacobiSym, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 261,
"column": 4
} | {
"line": 261,
"column": 23
} | {
"line": 262,
"column": 2
} | [
{
"pp": "a : ℤ\nh : 1 = 0\n⊢ False",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Int.instNeZeroOfNatOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring",
"instOfNatNat"... | [] | exact one_ne_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 477,
"column": 2
} | {
"line": 479,
"column": 95
} | {
"line": 481,
"column": 0
} | [
{
"pp": "q : ℕ\ninst✝ : NeZero q\na : ZMod q\nha : IsUnit a\n⊢ {p | Prime p ∧ ↑p = a}.Infinite",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real",
"Nat.Prime",
"instHDiv",
"ZMod.commRing",
"SummationFilter.instHasSupportOfLeAtTop",
"Real.instZero",
... | [] | by_contra! H
exact not_summable_residueClass_prime_div ha <|
summable_of_hasFiniteSupport <| show Set.Finite _ from support_residueClass_prime_div a ▸ H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 477,
"column": 2
} | {
"line": 479,
"column": 95
} | {
"line": 481,
"column": 0
} | [
{
"pp": "q : ℕ\ninst✝ : NeZero q\na : ZMod q\nha : IsUnit a\n⊢ {p | Prime p ∧ ↑p = a}.Infinite",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real",
"Nat.Prime",
"instHDiv",
"ZMod.commRing",
"SummationFilter.instHasSupportOfLeAtTop",
"Real.instZero",
... | [] | by_contra! H
exact not_summable_residueClass_prime_div ha <|
summable_of_hasFiniteSupport <| show Set.Finite _ from support_residueClass_prime_div a ▸ H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 307,
"column": 6
} | {
"line": 307,
"column": 16
} | {
"line": 307,
"column": 17
} | [
{
"pp": "a : ℤ\nR : Type u_1\ninst✝ : Semiring R\nχ : R →* ℤ\nhp : ∀ (p : ℕ) (pp : Nat.Prime p), p ≠ 2 → legendreSym p a = χ ↑p\nb : ℕ\nhb : Odd b\n⊢ J(a | b) = (List.map (⇑χ) (List.map Nat.cast b.primeFactorsList)).prod",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"a : ℤ\nR : Type u_1\ninst✝ : Semiring R\nχ : R →* ℤ\nhp : ∀ (p : ℕ) (pp : Nat.Prime p), p ≠ 2 → legendreSym p a = χ ↑p\nb : ℕ\nhb : Odd b\n⊢ (List.pmap (fun p pp ↦ legendreSym p a) b.primeFactorsList ⋯).prod =\n (List.map (⇑χ) (List.map Nat.cast b.primeFactorsList)).prod"
] | jacobiSym, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 79
} | {
"line": 204,
"column": 4
} | [
{
"pp": "case refine_1\ns T ε : ℝ\nS : ℝ → ℂ\nhs : 1 < s\nhS₁ : LocallyIntegrableOn (fun t ↦ S t) (Set.Ici 1) volume\nhS₂ : ∀ t ≥ T, ‖S t‖ ≤ ε * t\nh : LocallyIntegrableOn (fun t ↦ ‖S t‖ * t ^ (-s - 1)) (Set.Ici 1) volume\n⊢ ∀ᶠ (x : ℝ) in atTop, ‖‖S x‖ * x ^ (-s - 1)‖ ≤ ε * ‖x ^ (-s)‖",
"ppTerm": "?refine_1... | [
"s T ε : ℝ\nS : ℝ → ℂ\nhs : 1 < s\nhS₁ : LocallyIntegrableOn (fun t ↦ S t) (Set.Ici 1) volume\nhS₂ : ∀ t ≥ T, ‖S t‖ ≤ ε * t\nh : LocallyIntegrableOn (fun t ↦ ‖S t‖ * t ^ (-s - 1)) (Set.Ici 1) volume\nt : ℝ\nht : T ≤ t\nht' : 0 < t\n⊢ ‖‖S t‖ * t ^ (-s - 1)‖ ≤ ε * ‖t ^ (-s)‖"
] | filter_upwards [eventually_ge_atTop T, eventually_gt_atTop 0] with t ht ht' | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 164,
"column": 44
} | {
"line": 164,
"column": 83
} | {
"line": 164,
"column": 83
} | [
{
"pp": "p q : ℕ\nhp0 : ¬p = 0\nx : ℕ\nhx : x ∈ Ico 1 (p / 2).succ\n⊢ x * q / p ≤ p / 2 * q / p",
"ppTerm": "?m.194",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"instHDiv",
"Finset",
"Preorder.toLE",
"Nat.instLocallyFiniteOrder",
"Membership.mem",
... | [
"p q : ℕ\nhp0 : ¬p = 0\nx : ℕ\nhx : x ∈ Ico 1 (p / 2).succ\nthis : x ≤ p / 2\n⊢ x * q / p ≤ p / 2 * q / p"
] | have := le_of_lt_succ (mem_Ico.mp hx).2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 166,
"column": 15
} | {
"line": 166,
"column": 71
} | {
"line": 168,
"column": 0
} | [
{
"pp": "p q : ℕ\nhp0 : ¬p = 0\n⊢ ∑ a ∈ Ico 1 (p / 2).succ, #({x ∈ Ico 1 (q / 2).succ | x * p ≤ a * q}) =\n #({x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.2 * p ≤ x.1 * q})",
"ppTerm": "?m.211",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Finset.card_eq_s... | [] | by simp only [card_eq_sum_ones, sum_filter, sum_product] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 380,
"column": 19
} | {
"line": 380,
"column": 32
} | {
"line": 380,
"column": 33
} | [
{
"pp": "m₁ m₂ n : ℕ\n⊢ J(χ₄ ↑(m₁ * m₂) | n) = J(χ₄ ↑m₁ | n) * J(χ₄ ↑m₂ | n)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"ZMod.χ₄",
"ZMod.commRing",
"congrArg",
"CommSemiring.toS... | [
"m₁ m₂ n : ℕ\n⊢ J(χ₄ (↑m₁ * ↑m₂) | n) = J(χ₄ ↑m₁ | n) * J(χ₄ ↑m₂ | n)"
] | Nat.cast_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 223,
"column": 42
} | {
"line": 223,
"column": 57
} | {
"line": 223,
"column": 57
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nε : ℝ\nhε : 0 < ε\nh_lim' : Tendsto (fun t ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k) / ↑t) atTop (𝓝 l)\nt : ℝ\nht₁ : 0 < t\nht₂ : ‖∑ k ∈ Icc 1 ⌊t⌋₊, f k - l * ↑t‖ / t < ε\n⊢ ‖∑ k ∈ Icc 1 ⌊t⌋₊, f k - l * ↑t‖ < ε * t",
"ppTerm": "?m.... | [
"f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nε : ℝ\nhε : 0 < ε\nh_lim' : Tendsto (fun t ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k) / ↑t) atTop (𝓝 l)\nt : ℝ\nht₁ : 0 < t\nht₂ : ‖∑ k ∈ Icc 1 ⌊t⌋₊, f k - l * ↑t‖ < ε * t\n⊢ ‖∑ k ∈ Icc 1 ⌊t⌋₊, f k - l * ↑t‖ < ε * t"
] | div_lt_iff₀ ht₁ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 413,
"column": 40
} | {
"line": 413,
"column": 53
} | {
"line": 413,
"column": 54
} | [
{
"pp": "a✝ b : ℕ\nha : Odd a✝\nhb : Odd b\na x y : ℕ\n⊢ qrSign x a * qrSign y a * J(↑(x * y) | a) = qrSign x a * J(↑x | a) * (qrSign y a * J(↑y | a))",
"ppTerm": "?m.225",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
... | [
"a✝ b : ℕ\nha : Odd a✝\nhb : Odd b\na x y : ℕ\n⊢ qrSign x a * qrSign y a * J(↑x * ↑y | a) = qrSign x a * J(↑x | a) * (qrSign y a * J(↑y | a))"
] | Nat.cast_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 42
} | {
"line": 198,
"column": 6
} | [
{
"pp": "p q : ℕ\nhp : Fact (Nat.Prime p)\nhq0 : ↑q ≠ 0\nhswap :\n #({x ∈ Ico 1 (q / 2).succ ×ˢ Ico 1 (p / 2).succ | x.2 * q ≤ x.1 * p}) =\n #({x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.1 * q ≤ x.2 * p})\nhdisj :\n Disjoint ({x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.2 * p ≤ x.1 * q})\n ({... | [
"p q : ℕ\nhp : Fact (Nat.Prime p)\nhq0 : ↑q ≠ 0\nhswap :\n #({x ∈ Ico 1 (q / 2).succ ×ˢ Ico 1 (p / 2).succ | x.2 * q ≤ x.1 * p}) =\n #({x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.1 * q ≤ x.2 * p})\nhdisj :\n Disjoint ({x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.2 * p ≤ x.1 * q})\n ({x ∈ Ico 1 (p... | have := le_total (x.2 * p) (x.1 * q) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 192,
"column": 2
} | {
"line": 199,
"column": 11
} | {
"line": 200,
"column": 2
} | [
{
"pp": "p q : ℕ\nhp : Fact (Nat.Prime p)\nhq0 : ↑q ≠ 0\nhswap :\n #({x ∈ Ico 1 (q / 2).succ ×ˢ Ico 1 (p / 2).succ | x.2 * q ≤ x.1 * p}) =\n #({x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.1 * q ≤ x.2 * p})\nhdisj :\n Disjoint ({x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.2 * p ≤ x.1 * q})\n ({... | [
"p q : ℕ\nhp : Fact (Nat.Prime p)\nhq0 : ↑q ≠ 0\nhswap :\n #({x ∈ Ico 1 (q / 2).succ ×ˢ Ico 1 (p / 2).succ | x.2 * q ≤ x.1 * p}) =\n #({x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.1 * q ≤ x.2 * p})\nhdisj :\n Disjoint ({x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.2 * p ≤ x.1 * q})\n ({x ∈ Ico 1 (p... | have hunion :
{x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.2 * p ≤ x.1 * q} ∪
{x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.1 * q ≤ x.2 * p} =
Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ :=
Finset.ext fun x => by
have := le_total (x.2 * p) (x.1 * q)
simp only [mem_union, mem_fi... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 466,
"column": 6
} | {
"line": 466,
"column": 19
} | {
"line": 466,
"column": 20
} | [
{
"pp": "case inr\na b : ℕ\nhb : Odd b\nha₀ : a ≠ 0\nhb' : Odd (b % (4 * a))\ne a' : ℕ\nha₁' : ¬2 ∣ a'\nha₂ : a = 2 ^ e * a'\nha₁ : Odd a'\n⊢ J(↑(2 ^ e * a') | b) = J(↑(2 ^ e * a') | b % (4 * a))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCo... | [
"case inr\na b : ℕ\nhb : Odd b\nha₀ : a ≠ 0\nhb' : Odd (b % (4 * a))\ne a' : ℕ\nha₁' : ¬2 ∣ a'\nha₂ : a = 2 ^ e * a'\nha₁ : Odd a'\n⊢ J(↑(2 ^ e) * ↑a' | b) = J(↑(2 ^ e) * ↑a' | b % (4 * a))"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 7
} | {
"line": 502,
"column": 2
} | [
{
"pp": "p' : ℕ\nk : ℤ\nh : s p' = (2 ^ (p' + 2) - 1) * k\n⊢ ∃ k, ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1",
"ppTerm": "?m.170",
"assigned": true,
"usedConstants": [
"PNat.val",
"Int.cast",
"HMul.hMul",
"LucasLehmer.X.instMul",
"AddGroupWithOne.toAd... | [
"case h\np' : ℕ\nk : ℤ\nh : s p' = (2 ^ (p' + 2) - 1) * k\n⊢ ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.ModularForms.CongruenceSubgroups | {
"line": 262,
"column": 22
} | {
"line": 262,
"column": 35
} | {
"line": 262,
"column": 36
} | [
{
"pp": "g : GL (Fin 2) ℚ\nM : ℕ\ninst✝ : NeZero M\nA₁ : Matrix (Fin 2) (Fin 2) ℚ := ↑g\nA₂ : Matrix (Fin 2) (Fin 2) ℚ := ↑g⁻¹\nhA₁₂ : A₁ * A₂ = 1\na₁ : ℕ := A₁.den\na₂ : ℕ := A₂.den\nx✝ : SL(2, ℤ)\ny : Matrix (Fin 2) (Fin 2) ℤ\nhy : y.det = 1\nhy' : ⟨y, hy⟩ ∈ Γ(a₁ * a₂ * M)\nk : Matrix (Fin 2) (Fin 2) ℤ\nhk : ... | [
"g : GL (Fin 2) ℚ\nM : ℕ\ninst✝ : NeZero M\nA₁ : Matrix (Fin 2) (Fin 2) ℚ := ↑g\nA₂ : Matrix (Fin 2) (Fin 2) ℚ := ↑g⁻¹\nhA₁₂ : A₁ * A₂ = 1\na₁ : ℕ := A₁.den\na₂ : ℕ := A₂.den\nx✝ : SL(2, ℤ)\ny : Matrix (Fin 2) (Fin 2) ℤ\nhy : y.det = 1\nhy' : ⟨y, hy⟩ ∈ Γ(a₁ * a₂ * M)\nk : Matrix (Fin 2) (Fin 2) ℤ\nhk : y = 1 + (a₁ ... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.CongruenceSubgroups | {
"line": 262,
"column": 36
} | {
"line": 262,
"column": 49
} | {
"line": 262,
"column": 50
} | [
{
"pp": "g : GL (Fin 2) ℚ\nM : ℕ\ninst✝ : NeZero M\nA₁ : Matrix (Fin 2) (Fin 2) ℚ := ↑g\nA₂ : Matrix (Fin 2) (Fin 2) ℚ := ↑g⁻¹\nhA₁₂ : A₁ * A₂ = 1\na₁ : ℕ := A₁.den\na₂ : ℕ := A₂.den\nx✝ : SL(2, ℤ)\ny : Matrix (Fin 2) (Fin 2) ℤ\nhy : y.det = 1\nhy' : ⟨y, hy⟩ ∈ Γ(a₁ * a₂ * M)\nk : Matrix (Fin 2) (Fin 2) ℤ\nhk : ... | [
"g : GL (Fin 2) ℚ\nM : ℕ\ninst✝ : NeZero M\nA₁ : Matrix (Fin 2) (Fin 2) ℚ := ↑g\nA₂ : Matrix (Fin 2) (Fin 2) ℚ := ↑g⁻¹\nhA₁₂ : A₁ * A₂ = 1\na₁ : ℕ := A₁.den\na₂ : ℕ := A₂.den\nx✝ : SL(2, ℤ)\ny : Matrix (Fin 2) (Fin 2) ℤ\nhy : y.det = 1\nhy' : ⟨y, hy⟩ ∈ Γ(a₁ * a₂ * M)\nk : Matrix (Fin 2) (Fin 2) ℤ\nhk : y = 1 + (a₁ ... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.Basic | {
"line": 649,
"column": 92
} | {
"line": 650,
"column": 80
} | {
"line": 652,
"column": 0
} | [
{
"pp": "Γ : Subgroup (GL (Fin 2) ℝ)\ninst✝ : Γ.HasDetPlusMinusOne\nk : ℤ\nf : ModularForm Γ k\nn : ℕ\n⊢ (DirectSum.of (ModularForm Γ) k) f ^ n = (DirectSum.of (ModularForm Γ) (↑n * k)) (f.pow n)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.ModularFo... | [] | by
grind [DirectSum.ofPow, DirectSum.of_eq_of_gradedMonoid_eq (gnpow_eq_pow f n)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.Petersson | {
"line": 46,
"column": 2
} | {
"line": 57,
"column": 81
} | {
"line": 59,
"column": 0
} | [
{
"pp": "k : ℤ\nf f' : ℍ → ℂ\ng : GL (Fin 2) ℝ\nτ : ℍ\nD : ℝ := |↑(Matrix.GeneralLinearGroup.det g)|\nhD : ↑D ≠ 0\nj : ℂ := denom g ↑τ\n⊢ petersson k (f ∣[k] g) (f' ∣[k] g) τ = ↑D ^ (k - 2) * (σ g) (petersson k f f' (g • τ))",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"UpperHalfP... | [] | calc petersson k (f ∣[k] g) (f' ∣[k] g) τ
_ = D ^ (k - 2 + k) * conj (σ g (f (g • τ))) * σ g (f' (g • τ))
* (τ.im ^ k * j.normSq ^ (-k)) := by
simp [Complex.normSq_eq_conj_mul_self, (by abel : k - 2 + k = (k - 1) + (k - 1)), petersson,
zpow_add₀ hD, mul_zpow, ModularForm.slash_def, -Matrix.GeneralLine... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.NumberTheory.ModularForms.LevelOne.Basic | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 98
} | {
"line": 56,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nk : ℤ\ninst✝ : SlashInvariantFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nf : F\nc : ℂ\nhf : ⇑f = Function.const ℍ c\nthis : SlashInvariantFormClass F (Subgroup.map (Matrix.SpecialLinearGroup.mapGL ℝ) Γ(1)) k\nhI : c = ↑UpperHalfPlane.I ^ k * c\nh2I2 : ... | [] | simpa using ofReal_inj.trans <| zpow_eq_one_iff_right₀ (two_pos.le : (0 : ℝ) ≤ 2) (by norm_num1) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.IsBoundedAtImInfty | {
"line": 63,
"column": 63
} | {
"line": 63,
"column": 91
} | {
"line": 64,
"column": 4
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\na : Fin 2 → ZMod N\nk : ℤ\nhk : 3 ≤ k\nA : SL(2, ℤ)\nz : ℍ\nhz : 2 ≤ z.im\nn : ℤ\nhn : ModularGroup.T ^ (↑N * n) • z ∈ verticalStrip (↑N) z.im\n⊢ ‖eisensteinSeries (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) A)) k z‖ ≤\n ∑' (x : Fin 2 → ℤ), r { coe := { re :=... | [
"N : ℕ\ninst✝ : NeZero N\na : Fin 2 → ZMod N\nk : ℤ\nhk : 3 ≤ k\nA : SL(2, ℤ)\nz : ℍ\nhz : 2 ≤ z.im\nn : ℤ\nhn : ModularGroup.T ^ (↑N * n) • z ∈ verticalStrip (↑N) z.im\n⊢ ‖(eisensteinSeriesSIF (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) A)) k) z‖ ≤\n ∑' (x : Fin 2 → ℤ), r { coe := { re := ↑N, im... | ← eisensteinSeriesSIF_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.CuspFormSubmodule | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 63
} | {
"line": 120,
"column": 2
} | [
{
"pp": "k : ℤ\nf : ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nh : (PowerSeries.coeff 0) (qExpansion 1 ⇑f) = 0\nc : OnePoint ℝ\nhc : IsCusp c (Matrix.SpecialLinearGroup.mapGL ℝ).range\nγ : SL(2, ℤ)\na✝ : (Matrix.SpecialLinearGroup.mapGL ℝ) γ • ∞ = c\n⊢ IsZeroAtImInfty (⇑f ∣[k] γ)",
"ppTerm": "... | [
"k : ℤ\nf : ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nh : (PowerSeries.coeff 0) (qExpansion 1 ⇑f) = 0\nc : OnePoint ℝ\nhc : IsCusp c (Matrix.SpecialLinearGroup.mapGL ℝ).range\nγ : SL(2, ℤ)\na✝ : (Matrix.SpecialLinearGroup.mapGL ℝ) γ • ∞ = c\n⊢ IsZeroAtImInfty ⇑f"
] | rw [show (⇑f ∣[k] γ) = ⇑f from f.slash_action_eq' _ ⟨γ, rfl⟩] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 37
} | {
"line": 104,
"column": 0
} | [
{
"pp": "G✝ : Type u_1\ninst✝⁹ : Neg G✝\ninst✝⁸ : Preorder G✝\ninst✝⁷ : LocallyFiniteOrder G✝\ninst✝⁶ : atTop.NeBot\nG : Type u_2\ninst✝⁵ : AddCommGroup G\ninst✝⁴ : PartialOrder G\ninst✝³ : IsOrderedAddMonoid G\ninst✝² : LocallyFiniteOrder G\ninst✝¹ : NoTopOrder G\ninst✝ : NoBotOrder G\n⊢ Tendsto (fun g ↦ Ioo (... | [] | exact tendsto_Ioo_neg_atTop_atTop | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Modular | {
"line": 545,
"column": 4
} | {
"line": 546,
"column": 30
} | {
"line": 547,
"column": 2
} | [
{
"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 ^ ↑g 0 0 * S\n⊢ (S • z).re = -z.re",
"ppTerm": "?m.918",
"assigned": tru... | [] | rw [modular_S_smul, ← coe_re, coe_mk, inv_re, normSq_eq_norm_sq, norm_neg, hz', one_pow,
div_one, neg_re, coe_re] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 51
} | {
"line": 164,
"column": 2
} | [
{
"pp": "case hg\nz : ℍ\n⊢ Tendsto (fun x ↦ ∑ n ∈ Finset.Ico (-↑↑x) ↑↑x, ∑' (m : ℤ), (1 / (↑m * ↑z + ↑n) - 1 / (↑m * ↑z + ↑n + 1))) atTop\n (𝓝 (-2 * ↑π * I / ↑z))",
"ppTerm": "?hg",
"assigned": true,
"usedConstants": [
"EisensteinSeries.tendsto_tsum_one_div_linear_sub_succ_eq"
],
"... | [] | exact tendsto_tsum_one_div_linear_sub_succ_eq z | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ModularForms.Derivative | {
"line": 236,
"column": 4
} | {
"line": 236,
"column": 47
} | {
"line": 237,
"column": 2
} | [
{
"pp": "k : ℤ\nF : ℍ → ℂ\nhF : MDiff F\nγ : SL(2, ℤ)\nz : ℍ\nhLHS :\n (serreDerivative (↑k) F ∣[k + 2] γ) z =\n (D F ∣[k + 2] γ) z - ↑k * 12⁻¹ * ((EisensteinSeries.E2 ∣[2] γ) z * (F ∣[k] γ) z)\n⊢ D (F ∣[k] γ) z =\n (D F ∣[k + 2] γ) z -\n ↑k * (2 * ↑π * I)⁻¹ *\n (↑(↑γ 1 0) /\n de... | [] | simp [normalizedDerivOfComplex_SL_slash hF] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 65
} | {
"line": 158,
"column": 0
} | [
{
"pp": "z : ℍ\na b : ℤ\nthis : Summable fun x ↦ (↑b - ↑a) * ((↑↑x * ↑z + ↑a) * (↑↑x * ↑z + ↑b))⁻¹\nm : { x // x ∉ {0} }\n⊢ (↑b - ↑a) * ((↑↑m * ↑z + ↑a) * (↑↑m * ↑z + ↑b))⁻¹ = 1 / (↑↑m * ↑z + ↑a) - 1 / (↑↑m * ↑z + ↑b)",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"_private.Mathlib.... | [] | grind [one_div_linear_sub_one_div_linear_eq z a b m (by grind)] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 216,
"column": 59
} | {
"line": 218,
"column": 55
} | {
"line": 220,
"column": 0
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nz : ℍ\n⊢ Summable fun x ↦ eisSummand ↑k ![x.1, x.2] z",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Equiv.instEquivLike",
"congrArg",
"summable_eisSummand",
"List.ofFn",
"PseudoMetricS... | [] | by
refine (finTwoArrowEquiv ℤ).summable_iff.mp <| (summable_eisSummand hk z).congr (fun v ↦ ?_)
simp [show ![v 0, v 1] = v from List.ofFn_inj.mp rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 66,
"column": 39
} | {
"line": 66,
"column": 52
} | {
"line": 66,
"column": 53
} | [
{
"pp": "case refine_1\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\nthis : (↑π * I * ↑n ^ 2 * τ).re = -π * τ.im * ↑n ^ 2\nm : ℕ\nhm : n ^ 2 = ↑m\n⊢ rexp (-π * τ.im) ^ ↑(n ^ 2) = y ^ n ^ 2",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq... | [
"case refine_1\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\nthis : (↑π * I * ↑n ^ 2 * τ).re = -π * τ.im * ↑n ^ 2\nm : ℕ\nhm : n ^ 2 = ↑m\n⊢ rexp (-π * τ.im) ^ n ^ 2 = y ^ n ^ 2"
] | rpow_intCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 80
} | {
"line": 73,
"column": 8
} | [
{
"pp": "k : ℤ\nf : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nc : OnePoint ℝ\nhc : IsCusp c (Matrix.SpecialLinearGroup.mapGL ℝ).range\nγ : SL(2, ℤ)\na✝ : (Matrix.SpecialLinearGroup.mapGL ℝ) γ • ∞ = c\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
... | [
"k : ℤ\nf : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nc : OnePoint ℝ\nhc : IsCusp c (Matrix.SpecialLinearGroup.mapGL ℝ).range\nγ : SL(2, ℤ)\na✝ : (Matrix.SpecialLinearGroup.mapGL ℝ) γ • ∞ = c\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)"
] | rw [divByDiscriminant_slash_eq f γ, IsBoundedAtImInfty, BoundedAtFilter] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Multiplicity | {
"line": 212,
"column": 2
} | {
"line": 214,
"column": 53
} | {
"line": 216,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℤ\nhxy : ↑p ∣ x + y\nhx : ¬↑p ∣ x\nn : ℕ\nhn : Odd n\n⊢ emultiplicity (↑p) (x ^ n + y ^ n) = emultiplicity (↑p) (x + y) + emultiplicity p n",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Odd.neg_pow"... | [] | rw [← sub_neg_eq_add] at hxy
rw [← sub_neg_eq_add, ← sub_neg_eq_add, ← Odd.neg_pow hn]
exact Int.emultiplicity_pow_sub_pow hp hp1 hxy hx n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Multiplicity | {
"line": 212,
"column": 2
} | {
"line": 214,
"column": 53
} | {
"line": 216,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℤ\nhxy : ↑p ∣ x + y\nhx : ¬↑p ∣ x\nn : ℕ\nhn : Odd n\n⊢ emultiplicity (↑p) (x ^ n + y ^ n) = emultiplicity (↑p) (x + y) + emultiplicity p n",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Odd.neg_pow"... | [] | rw [← sub_neg_eq_add] at hxy
rw [← sub_neg_eq_add, ← sub_neg_eq_add, ← Odd.neg_pow hn]
exact Int.emultiplicity_pow_sub_pow hp hp1 hxy hx n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 263,
"column": 6
} | {
"line": 265,
"column": 25
} | {
"line": 266,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact : ∀ (i : ι), IsC... | [] | split_ifs with his
· exact hAcompact i his
· exact K_compact i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 263,
"column": 6
} | {
"line": 265,
"column": 25
} | {
"line": 266,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact : ∀ (i : ι), IsC... | [] | split_ifs with his
· exact hAcompact i his
· exact K_compact i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Multiplicity | {
"line": 330,
"column": 6
} | {
"line": 330,
"column": 36
} | {
"line": 330,
"column": 37
} | [
{
"pp": "x y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ emultiplicity 2 ((x ^ 2) ^ d - (y ^ 2) ^ d) + 1 =\n emultiplicity 2 (x + y) + emultiplicity 2 (x - y) + emultiplicity 2 ↑(2 * d)",
"ppTerm": "?m.212",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"x y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ emultiplicity 2 (x ^ 2 - y ^ 2) + emultiplicity 2 ↑d + 1 =\n emultiplicity 2 (x + y) + emultiplicity 2 (x - y) + emultiplicity 2 ↑(2 * d)",
"x y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\... | Int.two_pow_sub_pow' d hxy4 _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 42
} | {
"line": 394,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\nL : Type u_2\ninst✝² : Field L\nw : InfinitePlace L\nv : InfinitePlace K\ninst✝¹ : Algebra v.Completion w.Completion\nφ : w.Completion →+* ℂ\ninst✝ : ComplexEmbedding.LiesOver φ (extensionEmbedding v)\nx : v.Completion\n⊢ φ ((algebraMap v.Completion w.Completion) x) = (e... | [] | simp_all [liesOver_iff, RingHom.ext_iff] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 42
} | {
"line": 394,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\nL : Type u_2\ninst✝² : Field L\nw : InfinitePlace L\nv : InfinitePlace K\ninst✝¹ : Algebra v.Completion w.Completion\nφ : w.Completion →+* ℂ\ninst✝ : ComplexEmbedding.LiesOver φ (extensionEmbedding v)\nx : v.Completion\n⊢ φ ((algebraMap v.Completion w.Completion) x) = (e... | [] | simp_all [liesOver_iff, RingHom.ext_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 42
} | {
"line": 394,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\nL : Type u_2\ninst✝² : Field L\nw : InfinitePlace L\nv : InfinitePlace K\ninst✝¹ : Algebra v.Completion w.Completion\nφ : w.Completion →+* ℂ\ninst✝ : ComplexEmbedding.LiesOver φ (extensionEmbedding v)\nx : v.Completion\n⊢ φ ((algebraMap v.Completion w.Completion) x) = (e... | [] | simp_all [liesOver_iff, RingHom.ext_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 35
} | {
"line": 380,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nh : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w\na : mixedSpace K\nha : a ∈ A\n⊢ normAtComplexPlaces a ∈ normAtComplexPlaces '' A",
... | [] | exact Set.mem_image_of_mem _ ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 172,
"column": 90
} | {
"line": 179,
"column": 74
} | {
"line": 181,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\n⊢ regOfFamily u = |(of fun i ↦ (logEmbedding K) (Additive.ofMul (u ((equivFinRank K).symm i)))).det|",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_add... | [] | by
by_cases hu : IsMaxRank u
· rw [regOfFamily_of_isMaxRank hu, ZLattice.covolume_eq_det _
(((basisOfIsMaxRank hu).restrictScalars ℤ).reindex (equivFinRank K)), Basis.coe_reindex]
congr 3 with i
simp [basisOfIsMaxRank_apply hu]
· rw [regOfFamily_eq_zero hu, det_eq_zero_of_not_linearIndependent_rows,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 206,
"column": 37
} | {
"line": 206,
"column": 55
} | {
"line": 206,
"column": 56
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nw₁ w₂ : InfinitePlace K\ne₁ : { w // w ≠ w₁ } ≃ Fin (rank K)\ne₂ : { w // w ≠ w₂ } ≃ Fin (rank K)\nf : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w₁).symm e₁... | [
"case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nw₁ w₂ : InfinitePlace K\ne₁ : { w // w ≠ w₁ } ≃ Fin (rank K)\ne₂ : { w // w ≠ w₂ } ≃ Fin (rank K)\nf : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w₁).symm e₁.symm)\ng : ... | Int.abs_negOnePow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 234,
"column": 4
} | {
"line": 235,
"column": 83
} | {
"line": 237,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nhx : mixedEmbedding.norm x ≠ 0\nB : Module.Basis (Fin (rank K)) ℝ (logSpace K) := ⋯\n⊢ ∃ e, ↑e + logMap x ∈ ZSpan.fundamentalDomain B",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"NumberField.Units.b... | [] | obtain ⟨⟨e, h₁⟩, h₂, -⟩ := ZSpan.exist_unique_vadd_mem_fundamentalDomain B (logMap x)
exact ⟨⟨e, by rwa [← Module.Basis.ofZLatticeBasis_span ℝ (unitLattice K)]⟩, h₂⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 234,
"column": 4
} | {
"line": 235,
"column": 83
} | {
"line": 237,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nhx : mixedEmbedding.norm x ≠ 0\nB : Module.Basis (Fin (rank K)) ℝ (logSpace K) := ⋯\n⊢ ∃ e, ↑e + logMap x ∈ ZSpan.fundamentalDomain B",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"NumberField.Units.b... | [] | obtain ⟨⟨e, h₁⟩, h₂, -⟩ := ZSpan.exist_unique_vadd_mem_fundamentalDomain B (logMap x)
exact ⟨⟨e, by rwa [← Module.Basis.ofZLatticeBasis_span ℝ (unitLattice K)]⟩, h₂⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 363,
"column": 2
} | {
"line": 364,
"column": 64
} | {
"line": 366,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nhx : x ≠ 0\nw : { w // w ≠ w₀ }\n⊢ realSpaceToLogSpace (↑expMap.symm (normAtAllPlaces ((mixedEmbedding K) x))) w = logMap ((mixedEmbedding K) x) w",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"NumberField.Infin... | [] | simp_rw [realSpaceToLogSpace_apply, sum_expMap_symm_apply hx, expMap_symm_apply,
logMap, normAtPlace_apply, mul_sub, mul_assoc, norm_eq_norm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 339,
"column": 4
} | {
"line": 348,
"column": 80
} | {
"line": 349,
"column": 2
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu v : Fin (rank K) → (𝓞 K)ˣ\nhv : IsMaxRank v\nh : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K\nhu : IsMaxRank u\n⊢ regOfFamily u / regOfFamily v =\n ↑((Subgroup.closure (Set.range u) ⊔ tors... | [] | have : span ℤ (Set.range (basisOfIsMaxRank hu)) ≤ span ℤ (Set.range (basisOfIsMaxRank hv)) := by
rw [← toAddSubgroup_le, span_basisOfIsMaxRank hu, span_basisOfIsMaxRank hv,
← map_logEmbedding_sup_torsion (Subgroup.closure (Set.range u)).toAddSubgroup,
← map_logEmbedding_sup_torsion (Subgroup.closu... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 339,
"column": 4
} | {
"line": 348,
"column": 80
} | {
"line": 349,
"column": 2
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu v : Fin (rank K) → (𝓞 K)ˣ\nhv : IsMaxRank v\nh : Subgroup.closure (Set.range u) ⊔ torsion K ≤ Subgroup.closure (Set.range v) ⊔ torsion K\nhu : IsMaxRank u\n⊢ regOfFamily u / regOfFamily v =\n ↑((Subgroup.closure (Set.range u) ⊔ tors... | [] | have : span ℤ (Set.range (basisOfIsMaxRank hu)) ≤ span ℤ (Set.range (basisOfIsMaxRank hv)) := by
rw [← toAddSubgroup_le, span_basisOfIsMaxRank hu, span_basisOfIsMaxRank hv,
← map_logEmbedding_sup_torsion (Subgroup.closure (Set.range u)).toAddSubgroup,
← map_logEmbedding_sup_torsion (Subgroup.closu... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Ideal.KummerDedekind | {
"line": 198,
"column": 56
} | {
"line": 198,
"column": 68
} | {
"line": 198,
"column": 69
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nθ : 𝓞 K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : NumberField K\nhp : ¬p ∣ exponent θ\nQ : ℤ[X]\nhQ : Polynomial.map (Int.castRingHom (ZMod p)) Q ∈ monicFactorsMod θ p\n⊢ span {(algebraMap ℤ (𝓞 K)) ↑p} ⊔ span {(aeval θ) Q} = span {↑p, (aeval θ) Q}",
"ppTerm": "?... | [
"K : Type u_1\ninst✝² : Field K\nθ : 𝓞 K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : NumberField K\nhp : ¬p ∣ exponent θ\nQ : ℤ[X]\nhQ : Polynomial.map (Int.castRingHom (ZMod p)) Q ∈ monicFactorsMod θ p\n⊢ span {↑p} ⊔ span {(aeval θ) Q} = span {↑p, (aeval θ) Q}"
] | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Ideal.KummerDedekind | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 40
} | {
"line": 222,
"column": 40
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nθ : 𝓞 K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : NumberField K\nhp : ¬p ∣ exponent θ\nQ : ℤ[X]\nhQ : Polynomial.map (Int.castRingHom (ZMod p)) Q ∈ monicFactorsMod θ p\nthis✝ : (span {↑p, (aeval θ) Q}).IsMaximal\nthis : (span {↑p, (aeval θ) Q}).LiesOver (span {↑p})\n... | [
"K : Type u_1\ninst✝² : Field K\nθ : 𝓞 K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : NumberField K\nhp : ¬p ∣ exponent θ\nQ : ℤ[X]\nhQ : Polynomial.map (Int.castRingHom (ZMod p)) Q ∈ monicFactorsMod θ p\nthis✝ : (span {↑p, (aeval θ) Q}).IsMaximal\nthis : (span {↑p, (aeval θ) Q}).LiesOver (span {↑p})\n⊢ Module.fin... | ← finrank_quotient_span_eq_natDegree | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.ExistsRamified | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 53
} | {
"line": 34,
"column": 2
} | [
{
"pp": "K : Type u_1\n𝒪 : Type u_2\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : CommRing 𝒪\ninst✝¹ : Algebra 𝒪 K\ninst✝ : IsIntegralClosure 𝒪 ℤ K\nH : Module.finrank ℚ K ≠ 1\n⊢ ∃ p, Nat.Prime p ∧ ¬Algebra.IsUnramifiedIn 𝒪 (Ideal.span {↑p})",
"ppTerm": "?m.25",
"assigned": true,
"usedCon... | [
"K : Type u_1\n𝒪 : Type u_2\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : CommRing 𝒪\ninst✝¹ : Algebra 𝒪 K\ninst✝ : IsIntegralClosure 𝒪 ℤ K\nH : Module.finrank ℚ K ≠ 1\nthis : 0 < Module.finrank ℚ K\n⊢ ∃ p, Nat.Prime p ∧ ¬Algebra.IsUnramifiedIn 𝒪 (Ideal.span {↑p})"
] | have : 0 < Module.finrank ℚ K := Module.finrank_pos | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.ExistsRamified | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 70
} | {
"line": 79,
"column": 2
} | [
{
"pp": "K : Type u_1\n𝒪 : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : NumberField K\ninst✝³ : CommRing 𝒪\ninst✝² : Algebra 𝒪 K\ninst✝¹ : IsIntegralClosure 𝒪 ℤ K\ninst✝ : IsGalois ℚ K\nH : 1 < Module.finrank ℚ K\nthis✝ : IsDomain 𝒪\nthis : IsDedekindDomain 𝒪\n⊢ ∃ p, Nat.Prime p ∧ ∀ (P : Ideal 𝒪) (x : P.IsPrime)... | [
"K : Type u_1\n𝒪 : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : NumberField K\ninst✝³ : CommRing 𝒪\ninst✝² : Algebra 𝒪 K\ninst✝¹ : IsIntegralClosure 𝒪 ℤ K\ninst✝ : IsGalois ℚ K\nH : 1 < Module.finrank ℚ K\nthis✝¹ : IsDomain 𝒪\nthis✝ : IsDedekindDomain 𝒪\nthis : IsFractionRing 𝒪 K\n⊢ ∃ p, Nat.Prime p ∧ ∀ (P : Ideal �... | have := IsIntegralClosure.isFractionRing_of_finite_extension ℤ ℚ K 𝒪 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 51,
"column": 20
} | {
"line": 51,
"column": 33
} | {
"line": 51,
"column": 34
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\nhJ : ClassGroup.mk0 J = C⁻¹\nI : ↥(Ideal (𝓞 K))⁰\n⊢ ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C ↔ C = ClassGroup.mk0 I ∧ ↑(absNorm ↑J * absNorm ↑I) ≤ s * ↑(absNorm ↑J)",
"ppTerm": "?m.115",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\nhJ : ClassGroup.mk0 J = C⁻¹\nI : ↥(Ideal (𝓞 K))⁰\n⊢ ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C ↔ C = ClassGroup.mk0 I ∧ ↑(absNorm ↑J) * ↑(absNorm ↑I) ≤ s * ↑(absNorm ↑J)"
] | Nat.cast_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 99
} | {
"line": 72,
"column": 0
} | [
{
"pp": "case mpr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\nx✝ : mixedSpace K\n⊢ (x✝ ∈ fundamentalCone K ∧ ∃ a ∈ ↑J, (mixedEmbedding K) ↑a = x✝) ∧ mixedEmbedding.norm x✝ ≤ s →\n ∃ x,\n (x ∈ {x | (toMixed K) x ∈ fundamentalCone K ∧ mixedEm... | [] | rintro ⟨⟨hx₁, ⟨x, hx₂, rfl⟩⟩, hx₃⟩
exact ⟨(toMixed K).symm (mixedEmbedding K x), ⟨⟨hx₁, hx₃⟩, ⟨(x : K), by simp [hx₂], rfl⟩⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 99
} | {
"line": 72,
"column": 0
} | [
{
"pp": "case mpr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\nx✝ : mixedSpace K\n⊢ (x✝ ∈ fundamentalCone K ∧ ∃ a ∈ ↑J, (mixedEmbedding K) ↑a = x✝) ∧ mixedEmbedding.norm x✝ ≤ s →\n ∃ x,\n (x ∈ {x | (toMixed K) x ∈ fundamentalCone K ∧ mixedEm... | [] | rintro ⟨⟨hx₁, ⟨x, hx₂, rfl⟩⟩, hx₃⟩
exact ⟨(toMixed K).symm (mixedEmbedding K x), ⟨⟨hx₁, hx₃⟩, ⟨(x : K), by simp [hx₂], rfl⟩⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 102,
"column": 57
} | {
"line": 102,
"column": 70
} | {
"line": 102,
"column": 71
} | [
{
"pp": "case e'_3\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nh₁ :\n ∀ (s : ℝ),\n {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =\n ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}\nh₂ : {x | x ∈ fundam... | [
"case e'_3\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nh₁ :\n ∀ (s : ℝ),\n {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =\n ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}\nh₂ : {x | x ∈ fundamentalCone K ... | Nat.cast_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 30
} | {
"line": 309,
"column": 4
} | [
{
"pp": "m p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nP : Ideal (𝓞 K)\nhP₁ : P.IsPrime\nhP₂ : P.LiesOver 𝒑\ninst✝ : NeZero m\nhK : IsCyclotomicExtension {m} ℚ K\nhm : p.Coprime m\nζ : 𝓞 K := ⋯.toInteger\nh₁ : ¬p ∣ exponent ζ\nh₂ :\n Irreducible ↑((primesOverSpanE... | [
"m p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nP : Ideal (𝓞 K)\nhP₁ : P.IsPrime\nhP₂ : P.LiesOver 𝒑\ninst✝ : NeZero m\nhK : IsCyclotomicExtension {m} ℚ K\nhm : p.Coprime m\nζ : 𝓞 K := ⋯.toInteger\nh₁ : ¬p ∣ exponent ζ\nh₂ :\n Irreducible ↑((primesOverSpanEquivMonicFac... | refine dvd_trans h₂.2.2 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 403,
"column": 4
} | {
"line": 403,
"column": 44
} | {
"line": 403,
"column": 45
} | [
{
"pp": "n m p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhn : n = p ^ (k + 1) * m\nhm : ¬p ∣ m\nthis✝³ : IsAbelianGalois ℚ K\nthis✝² : NeZero m\nthis✝¹ : NeZero n\nζ : K := zeta n ℚ K\nhζ : IsPrimitiveRoot (zeta n ℚ K) n\nζₘ : ... | [
"n m p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhn : n = p ^ (k + 1) * m\nhm : ¬p ∣ m\nthis✝³ : IsAbelianGalois ℚ K\nthis✝² : NeZero m\nthis✝¹ : NeZero n\nζ : K := zeta n ℚ K\nhζ : IsPrimitiveRoot (zeta n ℚ K) n\nζₘ : K := ζ ^ p ^... | ramificationIdxIn_eq_of_not_dvd p Fₘ hm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 23
} | {
"line": 155,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis :\n Tendsto (fun x ↦ ↑(Nat.card { I // ↑(absNorm ↑I) ≤ x }) / x + x⁻¹) atTop\n (𝓝\n (2 ^ nrRealPlaces K * (2 * π) ^ nrComplexPlaces K * regulator K * ↑(classNumber K) /\n (↑(torsionOrder K) * √|↑(discr K)|) +\n 0))\n⊢... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis :\n Tendsto (fun x ↦ ↑(Nat.card { I // ↑(absNorm ↑I) ≤ x }) / x + x⁻¹) atTop\n (𝓝\n (2 ^ nrRealPlaces K * (2 * π) ^ nrComplexPlaces K * regulator K * ↑(classNumber K) /\n (↑(torsionOrder K) * √|↑(discr K)|)))\n⊢ Tendsto (fun s ↦ ↑(Nat.car... | rw [add_zero] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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