module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.SmoothNumbers | {
"line": 334,
"column": 2
} | {
"line": 335,
"column": 9
} | {
"line": 335,
"column": 10
} | [
{
"pp": "N : ℕ\n⊢ N.smoothNumbersᶜ \\ {0} ⊆ {n | N ≤ n}",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"setOf",
"Set.instSingletonSet",
"id",
"instOfNatNat",
"LE.le",
"instLENat",
"Set.instCom... | [
"N : ℕ\n⊢ (factoredNumbers (Finset.range N))ᶜ \\ {0} ⊆ {n | N ≤ n}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.ExpLog | {
"line": 28,
"column": 38
} | {
"line": 34,
"column": 31
} | {
"line": 36,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : α → ℂ\nhsum : Summable f\n⊢ Summable fun n ↦ log (1 - f n)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Complex.log_one",
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"Semiring.toModule",
"Complex.instNormedAddCommGroup"... | [] | by
have hg : DifferentiableAt ℂ (fun z ↦ log (1 - z)) 0 := by
have : 1 - 0 ∈ slitPlane := (sub_zero (1 : ℂ)).symm ▸ one_mem_slitPlane
fun_prop
have : (fun z ↦ log (1 - z)) =O[𝓝 0] id := by
simpa only [sub_zero, log_one] using! hg.isBigO_sub
exact this.comp_summable hsum | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 48
} | {
"line": 49,
"column": 49
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nhs : abscissaOfAbsConv f < ↑s.re\n⊢ ∃ a, LSeriesSummable f ↑a ∧ a < s.re",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\ns : ℂ\nhs : abscissaOfAbsConv f < ↑s.re\n⊢ ∃ a, LSeriesSummable f ↑a ∧ a < s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 56,
"column": 15
} | {
"line": 56,
"column": 26
} | {
"line": 56,
"column": 27
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nhs : abscissaOfAbsConv f < ↑s.re\nx : ℝ\nhx₁ : abscissaOfAbsConv f < ↑x\nhx₂ : ↑x < ↑s.re\n⊢ x < s.re",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\ns : ℂ\nhs : abscissaOfAbsConv f < ↑s.re\nx : ℝ\nhx₁ : abscissaOfAbsConv f < ↑x\nhx₂ : ↑x < ↑s.re\n⊢ x < s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 60,
"column": 16
} | {
"line": 60,
"column": 27
} | {
"line": 60,
"column": 28
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nh : LSeriesSummable f s\n⊢ ↑s.re ∈ Real.toEReal '' {x | LSeriesSummable f ↑x}",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Set.mem_image._simp_1",
"setOf",
"EReal",
"Membership.mem",
... | [
"f : ℕ → ℂ\ns : ℂ\nh : LSeriesSummable f s\n⊢ LSeriesSummable f ↑s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 66,
"column": 60
} | {
"line": 66,
"column": 89
} | {
"line": 66,
"column": 90
} | [
{
"pp": "f : ℕ → ℂ\nx : ℝ\nh : ∀ (y : ℝ), x < y → LSeriesSummable f ↑y\ny : EReal\nhy : y ∈ lowerBounds (Real.toEReal '' {x | LSeriesSummable f ↑x})\na : EReal\n⊢ ∀ (a : ℝ), LSeriesSummable f ↑a → y ≤ ↑a",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"f : ℕ → ℂ\nx : ℝ\nh : ∀ (y : ℝ), x < y → LSeriesSummable f ↑y\ny : EReal\nhy : y ∈ lowerBounds (Real.toEReal '' {x | LSeriesSummable f ↑x})\na : EReal\n⊢ ∀ (a : ℝ), LSeriesSummable f ↑a → y ≤ ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 110,
"column": 14
} | [
{
"pp": "f : ℕ → ℂ\nh : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C\n⊢ abscissaOfAbsConv f ≤ 1",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\nh : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C\n⊢ abscissaOfAbsConv f ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 110,
"column": 69
} | {
"line": 110,
"column": 80
} | {
"line": 110,
"column": 81
} | [
{
"pp": "f : ℕ → ℂ\nh : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C\n⊢ ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
... | [
"f : ℕ → ℂ\nh : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C\n⊢ ∃ C, ∀ (n : ℕ), ¬n = 0 → ‖f n‖ ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | {
"line": 116,
"column": 14
} | [
{
"pp": "f : ℕ → ℂ\nh : f =O[atTop] fun x ↦ 1\n⊢ abscissaOfAbsConv f ≤ 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\nh : f =O[atTop] fun x ↦ 1\n⊢ abscissaOfAbsConv f ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 116,
"column": 63
} | {
"line": 116,
"column": 74
} | {
"line": 116,
"column": 75
} | [
{
"pp": "f : ℕ → ℂ\nh : f =O[atTop] fun x ↦ 1\n⊢ f =O[atTop] fun n ↦ ↑n ^ 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Asymptotics.IsBigO",
... | [
"f : ℕ → ℂ\nh : f =O[atTop] fun x ↦ 1\n⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convergence | {
"line": 130,
"column": 21
} | {
"line": 130,
"column": 32
} | {
"line": 130,
"column": 33
} | [
{
"pp": "f : ℕ → ℝ\nx : ℝ\nh : (abscissaOfAbsConv fun x ↦ ↑(f x)) < ↑x\naux : term (fun x ↦ ↑(f x)) ↑x = fun n ↦ ↑(if n = 0 then 0 else f n / ↑n ^ x)\nthis : Summable fun x_1 ↦ if x_1 = 0 then 0 else f x_1 / ↑x_1 ^ x\nn : ℕ\nhn : n ∈ {0}ᶜ\n⊢ ¬n = 0",
"ppTerm": "?m.91",
"assigned": false,
"usedConsta... | [
"f : ℕ → ℝ\nx : ℝ\nh : (abscissaOfAbsConv fun x ↦ ↑(f x)) < ↑x\naux : term (fun x ↦ ↑(f x)) ↑x = fun n ↦ ↑(if n = 0 then 0 else f n / ↑n ^ x)\nthis : Summable fun x_1 ↦ if x_1 = 0 then 0 else f x_1 / ↑x_1 ^ x\nn : ℕ\nhn : n ∈ {0}ᶜ\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convolution | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 41
} | {
"line": 59,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Zero R\nf : ArithmeticFunction R\nn : ℕ\n⊢ (toArithmeticFunction ⇑f) n = f n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ite_eq_right_iff._simp_1",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"instOfNatNat",
"Arithmetic... | [] | simp +contextual [toArithmeticFunction] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.Convolution | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 49
} | {
"line": 85,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf g : ℕ → R\nn : ℕ\n⊢ (f ⍟ g) n = ∑ p ∈ n.divisorsAntidiagonal, f p.1 * g p.2",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"Nat.divisorsAntidiagonal",
"ArithmeticFunct... | [
"R : Type u_1\ninst✝ : Semiring R\nf g : ℕ → R\nn : ℕ\n⊢ (∑ x ∈ n.divisorsAntidiagonal, if x.2 = 0 then 0 else if x.1 = 0 then 0 else f x.1 * g x.2) =\n ∑ p ∈ n.divisorsAntidiagonal, f p.1 * g p.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Convolution | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 53
} | {
"line": 144,
"column": 54
} | [
{
"pp": "f g : ℕ → ℂ\ns a b : ℂ\nhf : LSeriesHasSum f s a\nhg : LSeriesHasSum g s b\nhsum : Summable fun x ↦ term f s x.1 * term g s x.2\n⊢ LSeriesHasSum (f ⍟ g) s (a * b)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"LS... | [
"f g : ℕ → ℂ\ns a b : ℂ\nhf : LSeriesHasSum f s a\nhg : LSeriesHasSum g s b\nhsum : Summable fun x ↦ term f s x.1 * term g s x.2\n⊢ HasSum (fun n ↦ ∑' (b : ↑((fun p ↦ p.1 * p.2) ⁻¹' {n})), term f s (↑b).1 * term g s (↑b).2) (a * b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Deriv | {
"line": 51,
"column": 39
} | {
"line": 52,
"column": 9
} | {
"line": 52,
"column": 10
} | [
{
"pp": "f : ℕ → ℂ\nn : ℕ\ns : ℂ\nhn : n ≠ 0\n⊢ HasDerivAt (fun z ↦ ↑n ^ (-z)) (-log ↑n * ↑n ^ (-s)) s",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NormedCommRi... | [
"f : ℕ → ℂ\nn : ℕ\ns : ℂ\nhn : n ≠ 0\n⊢ HasDerivAt (fun z ↦ ↑n ^ (-z)) (log ↑n * ↑n ^ (-s) * -1) s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Deriv | {
"line": 70,
"column": 32
} | {
"line": 70,
"column": 65
} | {
"line": 70,
"column": 65
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nh : abscissaOfAbsConv f < ↑s.re\nx : ℝ\nhxs : x < s.re\nhf : LSeriesSummable f ↑x\ny : ℝ\nhxy : x < y\nhys : y < s.re\nS : Set ℂ := {z | y < z.re}\nh₀ : Summable fun n ↦ ‖term f (↑x) n‖\nh₁ : ∀ (n : ℕ), DifferentiableOn ℂ (fun x ↦ term f x n) S\nh₂ : IsOpen S\nn : ℕ\nz : ℂ\nhz : z ∈ S... | [] | by simpa using! (hxy.trans hz).le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 48
} | {
"line": 213,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : Summable fun x ↦ ‖f x‖\nhf₀ : f 0 = 0\nthis :\n Tendsto (fun n ↦ ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ ∑' (e : ℕ), f (p ^ e)) i) ... | [
"R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : Summable fun x ↦ ‖f x‖\nhf₀ : f 0 = 0\nthis :\n Tendsto (fun n ↦ ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ ∑' (e : ℕ), f (p ^ e)) i) atTop\n (... | let F : ℕ → R := fun p ↦ ∑' (e : ℕ), f (p ^ e) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 290,
"column": 2
} | {
"line": 291,
"column": 9
} | {
"line": 291,
"column": 10
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\np : ℕ\nhp : Nat.Prime p\nhsum : Summable fun x ↦ ‖f x‖\n⊢ Summable fun a ↦ ‖f p ^ a‖",
"ppTerm": "?m.51",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\np : ℕ\nhp : Nat.Prime p\nhsum : Summable fun x ↦ ‖f x‖\n⊢ Summable fun a ↦ ‖f p ^ a‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 506,
"column": 6
} | {
"line": 522,
"column": 55
} | {
"line": 523,
"column": 2
} | [
{
"pp": "case h.inr.inl.inr\nξ : ℝ\nu : ℤ\nih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = ξ.convergent n\nleft✝ : IsCoprime u ↑1\nh₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u\nh₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹\nht : ξ < ↑u\n⊢ ∃ n, ↑u = ξ.convergent n",
"ppTerm": "?h.inr.inl.inr... | [] | replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
have hξ₁ : ⌊ξ⌋ = u - 1 := by
rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
· rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 506,
"column": 6
} | {
"line": 522,
"column": 55
} | {
"line": 523,
"column": 2
} | [
{
"pp": "case h.inr.inl.inr\nξ : ℝ\nu : ℤ\nih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = ξ.convergent n\nleft✝ : IsCoprime u ↑1\nh₁ : ↑1 = 1 → -(1 / 2) < ξ - ↑u\nh₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹\nht : ξ < ↑u\n⊢ ∃ n, ↑u = ξ.convergent n",
"ppTerm": "?h.inr.inl.inr... | [] | replace h₁ := lt_sub_iff_add_lt'.mp (h₁ rfl)
have hξ₁ : ⌊ξ⌋ = u - 1 := by
rw [floor_eq_iff, cast_sub, cast_one, sub_add_cancel]
exact ⟨(((sub_lt_sub_iff_left _).mpr one_half_lt_one).trans h₁).le, ht⟩
rcases eq_or_ne ξ ⌊ξ⌋ with Hξ | Hξ
· rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.Deriv | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 15
} | {
"line": 115,
"column": 16
} | [
{
"pp": "f : ℕ → ℂ\ns : ℝ\nhs : abscissaOfAbsConv (logMul f) < ↑s\nn : ℕ\nhn : max 1 ⌈Real.exp 1⌉₊ ≤ n\n⊢ 1 ≤ Real.log ↑n",
"ppTerm": "?m.85",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\ns : ℝ\nhs : abscissaOfAbsConv (logMul f) < ↑s\nn : ℕ\nhn : max 1 ⌈Real.exp 1⌉₊ ≤ n\n⊢ 1 ≤ Real.log ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 91
} | {
"line": 314,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →* F\nh : ∀ {p : ℕ}, Nat.Prime p → ‖f p‖ < 1\ns : Finset ℕ\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nH₁ : ∏ p ∈ s with Nat.Prime p, ∑' (n : ℕ), f (p ^ n) = ∏ p ∈ s with Nat.Prime p, (1 - f p)⁻¹\nH₂ : ∀ {p : ℕ}, Nat.Pri... | [] | exact H₁ ▸ summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum f.map_one hmul H₂ s | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 349,
"column": 2
} | {
"line": 350,
"column": 9
} | {
"line": 350,
"column": 10
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\nhsum : Summable fun x ↦ ‖f x‖\nH : (fun p ↦ (1 - f ↑p)⁻¹) = fun p ↦ ∑' (e : ℕ), f (↑p ^ e)\n⊢ HasProd (fun p ↦ (1 - f ↑p)⁻¹) (∑' (n : ℕ), f n)",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\nhsum : Summable fun x ↦ ‖f x‖\nH : (fun p ↦ (1 - f ↑p)⁻¹) = fun p ↦ ∑' (e : ℕ), f (↑p ^ e)\n⊢ HasProd (fun p ↦ ∑' (e : ℕ), f ↑p ^ e) (∑' (n : ℕ), f n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 28
} | {
"line": 377,
"column": 29
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\nhsum : Summable fun x ↦ ‖f x‖\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nthis :\n Tendsto (fun n ↦ ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ ∑' (e : ℕ), f (p ^ e)) i) atTop\n (𝓝 (∑' (n : ℕ), f n... | [
"F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\nhsum : Summable fun x ↦ ‖f x‖\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nthis :\n Tendsto (fun n ↦ ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ ∑' (e : ℕ), f (p ^ e)) i) atTop\n (𝓝 (∑' (n : ℕ), f n))\nH : ∀ (n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Positivity | {
"line": 63,
"column": 57
} | {
"line": 63,
"column": 85
} | {
"line": 63,
"column": 86
} | [
{
"pp": "a : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nx : ℝ\nhx : abscissaOfAbsConv a < ↑x\n⊢ abscissaOfAbsConv a < ↑(↑x).re",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"EReal",
"id",
"Complex.ofReal",
"Complex.r... | [
"a : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nx : ℝ\nhx : abscissaOfAbsConv a < ↑x\n⊢ abscissaOfAbsConv a < ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 541,
"column": 4
} | {
"line": 541,
"column": 74
} | {
"line": 542,
"column": 2
} | [
{
"pp": "case refine_1\nξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\n⊢ IsCoprime q.num ↑q.den",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.Coprime",
"Rat.reduced",
"Rat.num",
"Int.isCoprime_iff_nat_coprime",
"Rat.den",
... | [] | exact isCoprime_iff_nat_coprime.mpr (natAbs_natCast q.den ▸ q.reduced) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 541,
"column": 4
} | {
"line": 541,
"column": 74
} | {
"line": 542,
"column": 2
} | [
{
"pp": "case refine_1\nξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\n⊢ IsCoprime q.num ↑q.den",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.Coprime",
"Rat.reduced",
"Rat.num",
"Int.isCoprime_iff_nat_coprime",
"Rat.den",
... | [] | exact isCoprime_iff_nat_coprime.mpr (natAbs_natCast q.den ▸ q.reduced) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 541,
"column": 4
} | {
"line": 541,
"column": 74
} | {
"line": 542,
"column": 2
} | [
{
"pp": "case refine_1\nξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\n⊢ IsCoprime q.num ↑q.den",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Nat.Coprime",
"Rat.reduced",
"Rat.num",
"Int.isCoprime_iff_nat_coprime",
"Rat.den",
... | [] | exact isCoprime_iff_nat_coprime.mpr (natAbs_natCast q.den ▸ q.reduced) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.Positivity | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 54
} | {
"line": 77,
"column": 55
} | [
{
"pp": "a : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nf : ℂ → ℂ\nhf : Differentiable ℂ f\nx : ℝ\nhx : abscissaOfAbsConv a ≤ ↑x\nhf' : Set.EqOn f (LSeries a) {s | x < s.re}\ny : ℝ\nhxy : x < max x y + 1\nhxy' : abscissaOfAbsConv a < ↑(max x y) + 1\nhys : ↑(max x y) + 1 ∈ {s | x < s.re}\n⊢ 0 < f (↑(max x y) + 1)",
... | [
"a : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nf : ℂ → ℂ\nhf : Differentiable ℂ f\nx : ℝ\nhx : abscissaOfAbsConv a ≤ ↑x\nhf' : Set.EqOn f (LSeries a) {s | x < s.re}\ny : ℝ\nhxy : x < max x y + 1\nhxy' : abscissaOfAbsConv a < ↑(max x y) + 1\nhys : ↑(max x y) + 1 ∈ {s | x < s.re}\n⊢ 0 < LSeries a (↑(max x y) + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 549,
"column": 26
} | {
"line": 549,
"column": 36
} | {
"line": 549,
"column": 37
} | [
{
"pp": "ξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\nhq₀ : 0 < ↑q.den\nhq₁ : 0 < ↑q.den * (2 * ↑q.den - 1)\nhq₂ : 0 < 2 * (↑q.den * ↑q.den)\n⊢ 1 / (2 * ↑q.den ^ 2) < (↑q.den * (2 * ↑q.den - 1))⁻¹",
"ppTerm": "?m.194",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [
"ξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\nhq₀ : 0 < ↑q.den\nhq₁ : 0 < ↑q.den * (2 * ↑q.den - 1)\nhq₂ : 0 < 2 * (↑q.den * ↑q.den)\n⊢ 1 / (2 * ↑q.den ^ 2) < 1 / (↑q.den * (2 * ↑q.den - 1))"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.Positivity | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 83
} | {
"line": 81,
"column": 6
} | [
{
"pp": "case refine_1\na : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nf : ℂ → ℂ\nhf : Differentiable ℂ f\nx : ℝ\nhx : abscissaOfAbsConv a ≤ ↑x\nhf' : Set.EqOn f (LSeries a) {s | x < s.re}\ny : ℝ\nhxy : x < max x y + 1\nhxy' : abscissaOfAbsConv a < ↑(max x y) + 1\nhys : ↑(max x y) + 1 ∈ {s | x < s.re}\nhfx : 0 < f (↑(m... | [
"case refine_1\na : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nf : ℂ → ℂ\nhf : Differentiable ℂ f\nx : ℝ\nhx : abscissaOfAbsConv a ≤ ↑x\nhf' : Set.EqOn f (LSeries a) {s | x < s.re}\ny : ℝ\nhxy : x < max x y + 1\nhxy' : abscissaOfAbsConv a < ↑(max x y) + 1\nhys : ↑(max x y) + 1 ∈ {s | x < s.re}\nhfx : 0 < f (↑(max x y) + 1)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 57
} | {
"line": 56,
"column": 58
} | [
{
"pp": "t : ℝ\nht : 0 < t\n⊢ rexp (-π * t) < 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"C... | [
"t : ℝ\nht : 0 < t\n⊢ 0 < π * t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 25
} | {
"line": 121,
"column": 26
} | [
{
"pp": "case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nn : ℕ\n⊢ (↑n + a) ^ 0 * rexp (-π * (↑n + a ^ 2) * t) = rexp (-π * a ^ 2 * t) * rexp (-π * t) ^ n",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [
"case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nn : ℕ\n⊢ (↑n + a) ^ 0 * rexp (-π * (↑n + a ^ 2) * t) = rexp (-π * a ^ 2 * t) * rexp (↑n * (-π * t))"
] | ← Real.exp_nat_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 160,
"column": 32
} | {
"line": 160,
"column": 42
} | {
"line": 160,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nr C : ℝ\nhC :\n ∀ᶠ (x : ℝ) in 𝓝 0,\n x ∈ Ioi 0 → ‖((fun x ↦ P.g x - P.g₀) ∘ fun x ↦ x⁻¹) x‖ ≤ C * ‖((fun x ↦ x ^ (-(r + P.k))) ∘ fun x ↦ x⁻¹) x‖\nx : ℝ\nhC' : ‖P.g x⁻¹ - P.g₀‖ ≤ C * ‖x⁻¹ ^ (-(r + P.k))‖\nhx : 0... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nr C : ℝ\nhC :\n ∀ᶠ (x : ℝ) in 𝓝 0,\n x ∈ Ioi 0 → ‖((fun x ↦ P.g x - P.g₀) ∘ fun x ↦ x⁻¹) x‖ ≤ C * ‖((fun x ↦ x ^ (-(r + P.k))) ∘ fun x ↦ x⁻¹) x‖\nx : ℝ\nhx : 0 < x\nh_nv2 : ↑(x ^ P.k) ≠ 0\nh_nv : P.ε⁻¹ * ↑(x ^ P.k) ≠ 0\nhC... | ← one_div, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 168,
"column": 6
} | {
"line": 168,
"column": 45
} | {
"line": 168,
"column": 46
} | [
{
"pp": "a : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\n⊢ ‖rexp (-π * t)‖ < 1",
"ppTerm": "?m.153",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"Real.pi",
"HMul.hMul",
"congrArg",
"Real.instLT",
"id",
"Real.exp",
"Real.instOn... | [
"a : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\n⊢ rexp (-π * t) < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 171,
"column": 28
} | {
"line": 171,
"column": 47
} | {
"line": 171,
"column": 48
} | [
{
"pp": "case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nh0' : ‖rexp (-π * t)‖ < 1\nn : ℕ\n⊢ ↑n * rexp (-π * (↑n + a ^ 2) * t) = ↑n * rexp (-π * t) ^ n * rexp (-π * a ^ 2 * t)",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring"... | [
"case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nh0' : ‖rexp (-π * t)‖ < 1\nn : ℕ\n⊢ ↑n * rexp (-π * (↑n + a ^ 2) * t) = ↑n * rexp (↑n * (-π * t)) * rexp (-π * a ^ 2 * t)"
] | ← Real.exp_nat_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 176,
"column": 8
} | {
"line": 176,
"column": 27
} | {
"line": 176,
"column": 28
} | [
{
"pp": "case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nn : ℕ\n⊢ a * rexp (-π * (↑n + a ^ 2) * t) = a * rexp (-π * a ^ 2 * t) * rexp (-π * t) ^ n",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Re... | [
"case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nn : ℕ\n⊢ a * rexp (-π * (↑n + a ^ 2) * t) = a * rexp (-π * a ^ 2 * t) * rexp (↑n * (-π * t))"
] | ← Real.exp_nat_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 40
} | {
"line": 128,
"column": 41
} | [
{
"pp": "case refine_1\nz τ : ℂ\nhτ : 0 < τ.im\n⊢ ∀ (i : ℤ), ‖jacobiTheta₂_term i z τ‖ ≤ ↑|i| ^ 0 * rexp (-π * (τ.im * ↑i ^ 2 - 2 * |z.im| * ↑|i|))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCom... | [
"case refine_1\nz τ : ℂ\nhτ : 0 < τ.im\n⊢ ∀ (i : ℤ), ‖jacobiTheta₂_term i z τ‖ ≤ rexp (-π * (τ.im * ↑i ^ 2 - 2 * |z.im| * ↑|i|))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 61
} | {
"line": 189,
"column": 2
} | [
{
"pp": "a : ℝ\nha : 0 ≤ a\n⊢ (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1",
"ppTerm": "?m.340",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"one_pow",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"SeminormedRing.toNorm",
... | [] | simpa only [inv_pow, one_pow] using! isBigO_one_aux.pow 2 | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 154,
"column": 8
} | {
"line": 154,
"column": 84
} | {
"line": 154,
"column": 85
} | [
{
"pp": "case refine_2.inr.inr.inl\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nhz : z.im = 0\n⊢ (Summable fun x ↦ rexp (-(2 * π * ↑x * z.im))) → False",
"ppTerm": "?refine_2.inr.inr.inl",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"False",
"Real",
"Real.pi"... | [
"case refine_2.inr.inr.inl\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nhz : z.im = 0\n⊢ 1 = 0 → False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 202,
"column": 39
} | {
"line": 202,
"column": 72
} | {
"line": 202,
"column": 73
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\ns : ℂ\nstep1 : mellin (fun t ↦ P.g (1 / t)) (-s) = mellin P.g s\nstep2 : mellin (fun t ↦ ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = mellin P.g s\nstep3 : mellin (fun t ↦ P.ε • ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\ns : ℂ\nstep1 : mellin (fun t ↦ P.g (1 / t)) (-s) = mellin P.g s\nstep2 : mellin (fun t ↦ ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = mellin P.g s\nstep3 : mellin (fun t ↦ P.ε • ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = P.ε • mellin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 22
} | {
"line": 214,
"column": 23
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nhP : IsStrongFEPair P\nr : ℝ\n⊢ P.f =O[atTop] fun x ↦ x ^ r",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nhP : IsStrongFEPair P\nr : ℝ\n⊢ P.f =O[atTop] fun x ↦ x ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 13
} | {
"line": 218,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nhP : IsStrongFEPair P\nr : ℝ\n⊢ P.f =O[𝓝[>] 0] fun x ↦ x ^ r",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nhP : IsStrongFEPair P\nr : ℝ\n⊢ P.f =O[𝓝[>] 0] fun x ↦ x ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 76,
"column": 35
} | {
"line": 76,
"column": 46
} | {
"line": 76,
"column": 47
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\n⊢ ∀ t ∈ Ioi 0, HasSum (... | [
"ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\n⊢ ∀ (t : ℝ), 0 < t → HasSum (fun i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 42
} | {
"line": 88,
"column": 43
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\nthis : ∀ (i : ι), ‖a i‖... | [
"ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\nthis : ∀ (i : ι), ‖a i‖ / (π * q i)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 20
} | {
"line": 170,
"column": 21
} | [
{
"pp": "a t : ℝ\nht : 0 < t\nthis :\n ∀ (n : ℤ), cexp (-(↑π * (↑n + ↑a) ^ 2 * ↑t)) = cexp (-(↑π * ↑a ^ 2 * ↑t)) * jacobiTheta₂_term n (↑a * I * ↑t) (I * ↑t)\n⊢ HasSum (fun x ↦ ↑(rexp (-π * (↑x + a) ^ 2 * t))) (cexp (-↑π * ↑a ^ 2 * ↑t) * jacobiTheta₂ (↑a * I * ↑t) (I * ↑t))",
"ppTerm": "?m.120",
"assig... | [
"a t : ℝ\nht : 0 < t\nthis :\n ∀ (n : ℤ), cexp (-(↑π * (↑n + ↑a) ^ 2 * ↑t)) = cexp (-(↑π * ↑a ^ 2 * ↑t)) * jacobiTheta₂_term n (↑a * I * ↑t) (I * ↑t)\n⊢ HasSum (fun x ↦ cexp (-(↑π * ↑a ^ 2 * ↑t)) * jacobiTheta₂_term x (↑a * I * ↑t) (I * ↑t))\n (cexp (-(↑π * ↑a ^ 2 * ↑t)) * jacobiTheta₂ (↑a * I * ↑t) (I * ↑t))"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 20
} | {
"line": 180,
"column": 21
} | [
{
"pp": "a t : ℝ\nht : 0 < t\nthis : ∀ (n : ℤ), cexp (2 * ↑π * I * ↑a * ↑n) * cexp (-(↑π * ↑n ^ 2 * ↑t)) = jacobiTheta₂_term n (↑a) (I * ↑t)\n⊢ HasSum (fun n ↦ cexp (2 * ↑π * I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t))) (jacobiTheta₂ (↑a) (I * ↑t))",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants... | [
"a t : ℝ\nht : 0 < t\nthis : ∀ (n : ℤ), cexp (2 * ↑π * I * ↑a * ↑n) * cexp (-(↑π * ↑n ^ 2 * ↑t)) = jacobiTheta₂_term n (↑a) (I * ↑t)\n⊢ HasSum (fun n ↦ jacobiTheta₂_term n (↑a) (I * ↑t)) (jacobiTheta₂ (↑a) (I * ↑t))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 190,
"column": 4
} | {
"line": 191,
"column": 11
} | {
"line": 191,
"column": 12
} | [
{
"pp": "case pos\nt : ℝ\nht : 0 < t\nthis : (a : Prop) → Decidable a\nk : ℤ\n⊢ HasSum (fun n ↦ if ↑n + ↑k = 0 then 0 else rexp (-π * (↑n + ↑k) ^ 2 * t)) (evenKernel (↑↑k) t - 1)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"... | [
"case pos\nt : ℝ\nht : 0 < t\nthis : (a : Prop) → Decidable a\nk : ℤ\n⊢ HasSum (fun n ↦ if n = -k then 0 else rexp (-(π * ↑(n + k) ^ 2 * t))) (evenKernel (↑↑k) t - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 192,
"column": 37
} | {
"line": 192,
"column": 55
} | {
"line": 192,
"column": 56
} | [
{
"pp": "a t : ℝ\nht : 0 < t\nthis✝ : (a : Prop) → Decidable a\nh : ¬∃ n, ↑n = a\nthis : ∀ (n : ℤ), ↑n + a ≠ 0\n⊢ HasSum (fun n ↦ if ↑n + a = 0 then 0 else rexp (-π * (↑n + a) ^ 2 * t)) (evenKernel (↑a) t - 0)",
"ppTerm": "?m.137",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr... | [
"a t : ℝ\nht : 0 < t\nthis✝ : (a : Prop) → Decidable a\nh : ¬∃ n, ↑n = a\nthis : ∀ (n : ℤ), ↑n + a ≠ 0\n⊢ HasSum (fun n ↦ rexp (-(π * (↑n + a) ^ 2 * t))) (evenKernel (↑a) t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 192,
"column": 2
} | {
"line": 195,
"column": 49
} | {
"line": 197,
"column": 0
} | [
{
"pp": "case neg\na t : ℝ\nht : 0 < t\nthis : (a : Prop) → Decidable a\nh : ¬∃ n, ↑n = a\n⊢ HasSum (fun n ↦ if ↑n + a = 0 then 0 else rexp (-π * (↑n + a) ^ 2 * t)) (evenKernel (↑a) t - 0)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Add... | [] | · suffices ∀ (n : ℤ), n + a ≠ 0 by simpa [this] using hasSum_int_evenKernel a ht
contrapose! h
let ⟨n, hn⟩ := h
exact ⟨-n, by simpa [neg_eq_iff_add_eq_zero]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 13
} | {
"line": 200,
"column": 14
} | [
{
"pp": "a t : ℝ\nht : 0 < t\n⊢ HasSum (fun n ↦ if n = 0 then 0 else cexp (2 * ↑π * I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t)))\n (↑(cosKernel (↑a) t) - 1)",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [
"a t : ℝ\nht : 0 < t\n⊢ HasSum (fun n ↦ if n = 0 then 0 else cexp (2 * ↑π * I * ↑a * ↑n) * cexp (-(↑π * ↑n ^ 2 * ↑t)))\n (↑(cosKernel (↑a) t) - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 336,
"column": 37
} | {
"line": 336,
"column": 48
} | {
"line": 336,
"column": 49
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\n⊢ -1 < (s - 1).re",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\n⊢ 0 < s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 337,
"column": 43
} | {
"line": 337,
"column": 54
} | {
"line": 337,
"column": 55
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\nh_re1 : -1 < (s - 1).re\n⊢ -1 < (s - ↑P.k - 1).re",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_o... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\nh_re1 : -1 < (s - 1).re\n⊢ P.k < s.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 367,
"column": 29
} | {
"line": 368,
"column": 11
} | {
"line": 368,
"column": 12
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nT : ℝ\nhT : 0 < T\nhτ' : T < τ.im\nS : ℝ\nhz : |z.im| < S\nV : Set (ℂ × ℂ) := {u | |u.im| < S} ×ˢ {v | T < v.im}\nhVo : IsOpen V\nu : ℤ → ℝ := fun n ↦ 2 * π * ↑|n| * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))\n⊢ Summable u",
"ppTerm": "?m.208",
"assigned": true,
"usedCo... | [
"z τ : ℂ\nhτ : 0 < τ.im\nT : ℝ\nhT : 0 < T\nhτ' : T < τ.im\nS : ℝ\nhz : |z.im| < S\nV : Set (ℂ × ℂ) := {u | |u.im| < S} ×ˢ {v | T < v.im}\nhVo : IsOpen V\nu : ℤ → ℝ := fun n ↦ 2 * π * ↑|n| * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))\n⊢ Summable fun n ↦ 2 * (π * (↑|n| * rexp (-π * (T * ↑n ^ 2 - 2 * (S * ↑|n|)))))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 15
} | {
"line": 275,
"column": 16
} | [
{
"pp": "a : UnitAddCircle\nr p : ℝ\nhp : 0 < p\nhp' : (fun x ↦ cosKernel a x - 1) =O[atTop] fun x ↦ rexp (-p * x)\n⊢ (fun x ↦ (ofReal ∘ cosKernel a) x - 1) =O[atTop] fun x ↦ x ^ r",
"ppTerm": "?m.223",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"Complex.instNorme... | [
"a : UnitAddCircle\nr p : ℝ\nhp : 0 < p\nhp' : (fun x ↦ cosKernel a x - 1) =O[atTop] fun x ↦ rexp (-p * x)\n⊢ (fun x ↦ ↑(cosKernel a x) - 1) =O[atTop] fun x ↦ x ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 18
} | {
"line": 431,
"column": 2
} | [
{
"pp": "a b : UnitAddCircle\nthis :\n ∀ (s : ℂ),\n completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =\n completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -\n ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s\n⊢ DifferentiableAt ℂ (fun s ↦ completedHurwitzZetaEven a ... | [
"a b : UnitAddCircle\nthis :\n ∀ (s : ℂ),\n completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =\n completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -\n ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s\n⊢ DifferentiableAt ℂ\n (fun x ↦\n completedHurwitzZetaEven₀ a... | rw [funext this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 457,
"column": 10
} | {
"line": 457,
"column": 21
} | {
"line": 457,
"column": 22
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\n⊢ ↑P.k - 0 ≠ 0",
"ppTerm": "?m.242",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"Real.instZero",
"congrArg",
"sub_zero",
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\n⊢ ¬P.k = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 442,
"column": 6
} | {
"line": 442,
"column": 29
} | {
"line": 442,
"column": 30
} | [
{
"pp": "case refine_2.refine_1\na : UnitAddCircle\ns : ℂ\nhs : s ≠ 0\nhs' : s ≠ 1 ∨ a ≠ 0\nh : s ≠ 1\n⊢ s / 2 ≠ ↑(1 / 2)",
"ppTerm": "?refine_2.refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.i... | [
"case refine_2.refine_1\na : UnitAddCircle\ns : ℂ\nhs : s ≠ 0\nhs' : s ≠ 1 ∨ a ≠ 0\nh : s ≠ 1\n⊢ ¬s / 2 = 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 478,
"column": 4
} | {
"line": 478,
"column": 96
} | {
"line": 478,
"column": 97
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\n⊢ 0 < (-I * τ).re",
"ppTerm": "?m.159",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Complex.mul_re",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Real.instZero",
... | [
"z τ : ℂ\nhτ : 0 < τ.im\n⊢ 0 < τ.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | {
"line": 147,
"column": 14
} | [
{
"pp": "x : ℝ\n⊢ oddKernel 0 x = 0",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\n⊢ oddKernel 0 x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 13
} | {
"line": 156,
"column": 14
} | [
{
"pp": "x : ℝ\n⊢ sinKernel 0 x = 0",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\n⊢ sinKernel 0 x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 39
} | {
"line": 102,
"column": 40
} | [
{
"pp": "s : ℂ\n⊢ completedRiemannZeta₀ (1 - s) = completedRiemannZeta₀ s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HurwitzZeta.completedHurwitzZetaEven₀",
"congrArg",
"HSub.hSub",
"AddCommGroup.toAddGroup",
"id",
"Sub... | [
"s : ℂ\n⊢ completedHurwitzZetaEven₀ 0 (1 - s) = completedRiemannZeta₀ s"
] | ← completedHurwitzZetaEven₀_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 49
} | {
"line": 131,
"column": 50
} | [
{
"pp": "s : ℂ\n⊢ hurwitzZeta 0 s = riemannZeta s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"riemannZeta",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Hurwit... | [
"s : ℂ\n⊢ hurwitzZetaOdd 0 s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.SumPrimeReciprocals | {
"line": 85,
"column": 17
} | {
"line": 85,
"column": 47
} | {
"line": 85,
"column": 48
} | [
{
"pp": "h : Summable ({p | Nat.Prime p}.indicator fun n ↦ 1 / ↑n)\nk : ℕ\nhk : ∑' (x : ℕ), ({p | Nat.Prime p} ∩ {p | k ≤ p}).indicator (fun n ↦ 1 / ↑n) x < 1 / 2\nh' : Summable (({p | Nat.Prime p} ∩ {p | k ≤ p}).indicator fun n ↦ 1 / ↑n)\np : ℕ\nhp : p ∈ (4 ^ (k.primesBelow.card + 1)).succ.primesBelow \\ k.pri... | [
"h : Summable ({p | Nat.Prime p}.indicator fun n ↦ 1 / ↑n)\nk : ℕ\nhk : ∑' (x : ℕ), ({p | Nat.Prime p} ∩ {p | k ≤ p}).indicator (fun n ↦ 1 / ↑n) x < 1 / 2\nh' : Summable (({p | Nat.Prime p} ∩ {p | k ≤ p}).indicator fun n ↦ 1 / ↑n)\np : ℕ\nhp : p ∈ (4 ^ (k.primesBelow.card + 1)).succ.primesBelow \\ k.primesBelow\nhp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 21
} | {
"line": 210,
"column": 22
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\n⊢ riemannZeta s = ∑' (n : ℕ), 1 / ↑n ^ s",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : ℂ\nhs : 1 < s.re\n⊢ riemannZeta s = ∑' (n : ℕ), 1 / ↑n ^ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 578,
"column": 42
} | {
"line": 578,
"column": 52
} | {
"line": 578,
"column": 53
} | [
{
"pp": "Λ : ℂ → ℂ\nhf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s\nL : ℂ\nh_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)\nclaim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t\nclaim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))\nhs' : 0 ≠ 1\nS_nhds : {1}ᶜ ∈ ... | [
"Λ : ℂ → ℂ\nhf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s\nL : ℂ\nh_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)\nclaim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t\nclaim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))\nhs' : 0 ≠ 1\nS_nhds : {1}ᶜ ∈ 𝓝 0\nx : ℂ\... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 24
} | {
"line": 592,
"column": 25
} | [
{
"pp": "case inr\na : UnitAddCircle\nh : a ≠ 0 ∨ 0 ≠ 0\n⊢ Function.update (fun s ↦ completedHurwitzZetaEven a s / s.Gammaℝ) 0 (if a = 0 then -1 / 2 else 0) 0 =\n completedHurwitzZetaEven a 0 / Gammaℝ 0",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.... | [
"case inr\na : UnitAddCircle\nh : a ≠ 0 ∨ 0 ≠ 0\n⊢ ¬a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 650,
"column": 2
} | {
"line": 650,
"column": 13
} | {
"line": 650,
"column": 14
} | [
{
"pp": "a : UnitAddCircle\n⊢ Tendsto (fun s ↦ hurwitzZetaEven a s - 1 / (s - 1) / s.Gammaℝ) (𝓝 1) (𝓝 (hurwitzZetaEven a 1))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.toInv",
"instHDiv",
"c... | [
"a : UnitAddCircle\n⊢ Tendsto (fun s ↦ hurwitzZetaEven a s - (s - 1)⁻¹ / s.Gammaℝ) (𝓝 1) (𝓝 (hurwitzZetaEven a 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 79,
"column": 2
} | {
"line": 80,
"column": 9
} | {
"line": 80,
"column": 10
} | [
{
"pp": "⊢ (abscissaOfAbsConv fun n ↦ ↑(μ n)) = 1",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"instInfSetEReal",
"Real",
"Set.Ioi",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"_private.Mathlib.NumberTheory.LSeri... | [
"⊢ sInf (Set.Ioo 1 ⊤) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 57
} | {
"line": 152,
"column": 58
} | [
{
"pp": "n : ℕ\nχ : DirichletCharacter ℂ n\nthis : (1 ⍟ fun x ↦ ↑(μ x)) = δ\n⊢ (fun n_1 ↦ χ ↑n_1) * 1 ⍟ ((fun n_1 ↦ χ ↑n_1) * fun n ↦ ↑(μ n)) = δ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"ArithmeticFunc... | [
"n : ℕ\nχ : DirichletCharacter ℂ n\nthis : (1 ⍟ fun x ↦ ↑(μ x)) = δ\n⊢ (fun x ↦ χ ↑x) * δ = δ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 9
} | {
"line": 203,
"column": 10
} | [
{
"pp": "N : ℕ\nhn : N ≠ 0\nχ : DirichletCharacter ℂ N\n⊢ (abscissaOfAbsConv fun n ↦ χ ↑n) = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instInfSetEReal",
"Real",
"Set.Ioi",
"ZMod.commRing",
"congrArg",
"PartialOrder.toPreorder",
... | [
"N : ℕ\nhn : N ≠ 0\nχ : DirichletCharacter ℂ N\n⊢ sInf (Set.Ioo 1 ⊤) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 13
} | {
"line": 211,
"column": 14
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\nf : ℕ → ℂ\ns : ℂ\nh : LSeriesSummable f s\nn : ℕ\n⊢ ‖((fun n ↦ χ ↑n) * f) n‖ ≤ ‖f n‖",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"ZMod.commRing",
... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\nf : ℕ → ℂ\ns : ℂ\nh : LSeriesSummable f s\nn : ℕ\n⊢ ‖χ ↑n‖ * ‖f n‖ ≤ ‖f n‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 230,
"column": 13
} | {
"line": 230,
"column": 28
} | {
"line": 230,
"column": 29
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nh : L (fun n ↦ χ ↑n) s = 0\n⊢ False",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nh : L (fun n ↦ χ ↑n) s = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 299,
"column": 13
} | {
"line": 299,
"column": 41
} | {
"line": 299,
"column": 42
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\nh : L (fun n ↦ ↑(ζ n)) s = 0\n⊢ False",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : ℂ\nhs : 1 < s.re\nh : L (fun n ↦ ↑(ζ n)) s = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 337,
"column": 34
} | {
"line": 337,
"column": 45
} | {
"line": 337,
"column": 46
} | [
{
"pp": "x : ℝ\nhx : 1 < x\n⊢ 1 < (↑x).re",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Real",
"Real.instLT",
"id",
"Complex.ofReal",
"Complex.re",
"Real.instOne",
"LT.lt",
"One.toOfNat1",
"OfNat.ofNat"
],
"usedFVars": [
... | [
"x : ℝ\nhx : 1 < x\n⊢ 1 < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 45
} | {
"line": 340,
"column": 46
} | [
{
"pp": "x : ℝ\nhx : 1 < x\nhx' : 1 < (↑x).re\n⊢ abscissaOfAbsConv 1 < ↑x",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"LSeries.abscissaOfAbsConv_one",
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"id",
"Pi.in... | [
"x : ℝ\nhx : 1 < x\nhx' : 1 < (↑x).re\n⊢ 1 < ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 372,
"column": 2
} | {
"line": 373,
"column": 9
} | {
"line": 373,
"column": 10
} | [
{
"pp": "n : ℕ\n⊢ ((fun n ↦ ↑(Λ n)) ⍟ fun n ↦ ↑(ζ n)) n = Complex.log ↑n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"CharP.cast_eq_zero",
"Eq.mpr",
"Complex.log",
"Nat.instMulZeroClass",
"Real",
"HMul.hMul",
... | [
"n : ℕ\n⊢ (∑ x ∈ n.divisorsAntidiagonal, if x.2 = 0 then 0 else ↑(Λ x.1)) = Complex.log ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 385,
"column": 4
} | {
"line": 386,
"column": 11
} | {
"line": 386,
"column": 12
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\nhf : Summable fun x ↦ ‖term (logMul 1) s x‖\nn : ℕ\n⊢ ‖(fun n ↦ ↑(Λ n)) n‖ ≤ ‖Complex.log ↑n‖",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"Norm.norm",
"Eq.mpr",
"Complex.log",
"Real.instLE",
... | [
"s : ℂ\nhs : 1 < s.re\nhf : Summable fun x ↦ ‖term (logMul 1) s x‖\nn : ℕ\n⊢ Λ n ≤ Real.log ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 40,
"column": 4
} | {
"line": 41,
"column": 11
} | {
"line": 41,
"column": 12
} | [
{
"pp": "s : ℂ\nhs : s ≠ 0\nm n : ℕ\n⊢ ↑(m * n) ^ (-s) = ↑m ^ (-s) * ↑n ^ (-s)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroOneClass",
"HMul.hMul",
"congrArg",
"Complex.instPow",
... | [
"s : ℂ\nhs : s ≠ 0\nm n : ℕ\n⊢ (↑m * ↑n) ^ (-s) = ↑m ^ (-s) * ↑n ^ (-s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 52,
"column": 4
} | {
"line": 54,
"column": 11
} | {
"line": 54,
"column": 12
} | [
{
"pp": "s : ℂ\nn✝ : ℕ\nχ : DirichletCharacter ℂ n✝\nhs : s ≠ 0\nm n : ℕ\n⊢ χ ↑(m * n) * ↑↑(m * n) ^ (-s) = χ ↑m * ↑↑m ^ (-s) * (χ ↑n * ↑↑n ^ (-s))",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne... | [
"s : ℂ\nn✝ : ℕ\nχ : DirichletCharacter ℂ n✝\nhs : s ≠ 0\nm n : ℕ\n⊢ χ ↑m * χ ↑n * (↑↑m ^ (-s) * ↑↑n ^ (-s)) = χ ↑m * ↑↑m ^ (-s) * (χ ↑n * ↑↑n ^ (-s))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 272,
"column": 2
} | {
"line": 273,
"column": 66
} | {
"line": 274,
"column": 4
} | [
{
"pp": "b p : ℝ\nhp : 0 < p\nhp' : HurwitzKernelBounds.F_int 1 ↑b =O[atTop] fun t ↦ rexp (-p * t)\nt : ℝ\nht : 0 < t\n⊢ ‖oddKernel (↑b) t‖ ≤ HurwitzKernelBounds.F_int 1 (↑b) t",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"HurwitzKernelBounds.f_int",
"Norm.norm",
"Int.... | [
"b p : ℝ\nhp : 0 < p\nhp' : HurwitzKernelBounds.F_int 1 ↑b =O[atTop] fun t ↦ rexp (-p * t)\nt : ℝ\nht : 0 < t\n⊢ |∑' (b_1 : ℤ), (↑b_1 + b) * rexp (-(π * (↑b_1 + b) ^ 2 * t))| ≤ ∑' (n : ℤ), |↑n + b| * rexp (-(π * (↑n + b) ^ 2 * t))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 15
} | {
"line": 318,
"column": 16
} | [
{
"pp": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x ↦ ‖oddKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)\n⊢ (fun x ↦ ‖(ofReal ∘ oddKernel a) x - 0‖) =O[atTop] fun x ↦ x ^ r",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toS... | [
"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x ↦ ‖oddKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)\n⊢ oddKernel a =O[atTop] fun x ↦ x ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 15
} | {
"line": 322,
"column": 16
} | [
{
"pp": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x ↦ ‖sinKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)\n⊢ (fun x ↦ ‖(ofReal ∘ sinKernel a) x - 0‖) =O[atTop] fun x ↦ x ^ r",
"ppTerm": "?m.223",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toS... | [
"a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x ↦ ‖sinKernel a x‖) =O[atTop] fun x ↦ rexp (-v * x)\n⊢ sinKernel a =O[atTop] fun x ↦ x ^ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 765,
"column": 4
} | {
"line": 765,
"column": 47
} | {
"line": 765,
"column": 48
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\n⊢ a ≠ 0 ∨ 1 - s ≠ 0",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"AddGroupWithOne.toAddGroup",
"congrArg",
"HSub.hSub",
"Complex.instZero",
"AddCo... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\n⊢ ¬a = 0 ∨ ¬1 = s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 767,
"column": 30
} | {
"line": 767,
"column": 41
} | {
"line": 767,
"column": 42
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\nthis : hurwitzZetaEven a (1 - s) = completedHurwitzZetaEven a (1 - s) * (1 - s).Gammaℝ⁻¹\n⊢ s ≠ 0",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\nthis : hurwitzZetaEven a (1 - s) = completedHurwitzZetaEven a (1 - s) * (1 - s).Gammaℝ⁻¹\n⊢ ¬s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 778,
"column": 4
} | {
"line": 778,
"column": 29
} | {
"line": 778,
"column": 30
} | [
{
"pp": "case h\na : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\n⊢ 1 - s ≠ 0",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"HSub.hSub",
"Complex.instZero",
"Complex.addGroupWithOne",
"id",
... | [
"case h\na : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\n⊢ ¬1 = s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 779,
"column": 69
} | {
"line": 779,
"column": 80
} | {
"line": 779,
"column": 81
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nthis : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹\nn : ℕ\n⊢ s ≠ -↑n",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"Complex.instNatCast",
"Nat.cast",
"Complex",... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nthis : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹\nn : ℕ\n⊢ ¬s = -↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 780,
"column": 48
} | {
"line": 780,
"column": 59
} | {
"line": 780,
"column": 60
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nthis : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹\n⊢ s ≠ 0",
"ppTerm": "?m.146",
"assigned": true,
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"Complex",
... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nthis : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹\n⊢ ¬s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Basic | {
"line": 231,
"column": 30
} | {
"line": 231,
"column": 46
} | {
"line": 231,
"column": 46
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\ninst✝ : NormalizedGCDMonoid R\nhn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n\na b c : R\ns : Finset R := {a, b, c}\nd : R := s.gcd id\nA : R\nhA : a = d * A\nB : R\nhB :... | [
"n : ℕ\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\ninst✝ : NormalizedGCDMonoid R\nhn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n\na b c : R\ns : Finset R := {a, b, c}\nd : R := s.gcd id\nA : R\nhA : a = d * A\nB : R\nhB : b = d * B\n... | normalize_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 13
} | {
"line": 38,
"column": 14
} | [
{
"pp": "z : ℤ\n⊢ ¬↑(z * z) = ↑2",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"NonUnitalNonAss... | [
"z : ℤ\n⊢ ¬↑z * ↑z = 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 148,
"column": 6
} | {
"line": 149,
"column": 26
} | {
"line": 149,
"column": 27
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n⊢ z = 0",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n⊢ z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 163,
"column": 6
} | {
"line": 164,
"column": 26
} | {
"line": 164,
"column": 27
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n⊢ z = 0",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n⊢ z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 32
} | {
"line": 165,
"column": 33
} | [
{
"pp": "case pos\nx y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\nhz : z = 0\n⊢ PythagoreanTriple (x / ↑(x.gcd y)) (y / ↑(x.gcd y)) (z / ↑(x.gcd y))",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Int.gcd",
"Eq.mpr... | [
"case pos\nx y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\nhz : z = 0\n⊢ PythagoreanTriple 0 0 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.FLT.Four | {
"line": 148,
"column": 17
} | {
"line": 148,
"column": 48
} | {
"line": 150,
"column": 0
} | [
{
"pp": "r s : ℤ\nh : IsCoprime r s\n⊢ IsCoprime (s ^ 2 + r ^ 2) s",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Int.isCoprime_of_sq_sum"
],
"usedFVars": [
"s",
"r",
"h"
],
"usedGoals": []
}
] | [] | apply Int.isCoprime_of_sq_sum h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 251,
"column": 6
} | {
"line": 251,
"column": 56
} | {
"line": 251,
"column": 57
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nhk : ∀ (x : K), 1 + x ^ 2 ≠ 0\nx : K\n⊢ ¬1 = -1",
"ppTerm": "?m.267",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.... | [
"K : Type u_1\ninst✝ : Field K\nhk : ∀ (x : K), 1 + x ^ 2 ≠ 0\nx : K\n⊢ ¬1 + 1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Radical.Basic | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 27
} | {
"line": 202,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nha : Prime a\nn : ℕ\nhn : n ≠ 0\n⊢ radical a = normalize a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.radical_of_prime"
... | [] | exact radical_of_prime ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Radical.Basic | {
"line": 220,
"column": 54
} | {
"line": 220,
"column": 84
} | {
"line": 220,
"column": 85
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nha : Irreducible a\nhb : b ≠ 0\nha' : a ∣ b\nc : M\nhc : c ∈ normalizedFactors b\nhc' : Associated a c\n⊢ c ∈ primeFactors b",
"ppTerm": "?m.73",
"assigned": true,
"use... | [
"M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nha : Irreducible a\nhb : b ≠ 0\nha' : a ∣ b\nc : M\nhc : c ∈ normalizedFactors b\nhc' : Associated a c\n⊢ c ∈ normalizedFactors b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 297,
"column": 2
} | {
"line": 299,
"column": 8
} | {
"line": 300,
"column": 2
} | [
{
"pp": "m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ m ^ 2 + n ^ 2\nh2m : ↑p ∣ 2 * m ^ 2\n⊢ False",
"ppTerm": "?m.146",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic... | [
"m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ m ^ 2 + n ^ 2\nh2m : ↑p ∣ 2 * m ^ 2\nh2n : ↑p ∣ 2 * n ^ 2\n⊢ False"
] | have h2n : (p : ℤ) ∣ 2 * n ^ 2 := by
convert! dvd_sub hp2 hp1 using 1
ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.FLT.MasonStothers | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 28
} | {
"line": 77,
"column": 29
} | [
{
"pp": "k : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b c : k[X]\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a b\nhsum : b + c + a = 0\nw : k[X] := a.wronskian b\nwab : w = a.wronskian b\nhbc : IsCoprime b c\nhsum' : b + c + a = 0\nhca : IsCoprime c a\nwbc : w = b.wronskian c\n⊢ w = c.wrons... | [
"k : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b c : k[X]\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a b\nhsum : b + c + a = 0\nw : k[X] := a.wronskian b\nwab : w = a.wronskian b\nhbc : IsCoprime b c\nhsum' : b + c + a = 0\nhca : IsCoprime c a\nwbc : w = b.wronskian c\n⊢ w = c.wronskian a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 472,
"column": 2
} | {
"line": 476,
"column": 7
} | {
"line": 477,
"column": 2
} | [
{
"pp": "case neg.inl.inl\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (... | [
"case neg.inl.inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (circleEquivG... | · -- m even, n even
exfalso
have h1 : 2 ∣ (Int.gcd n m : ℤ) :=
Int.dvd_coe_gcd (Int.dvd_of_emod_eq_zero hn2) (Int.dvd_of_emod_eq_zero hm2)
lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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