module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.FLT.Three | {
"line": 714,
"column": 4
} | {
"line": 714,
"column": 19
} | {
"line": 714,
"column": 20
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nh : S.multiplicity = 1\n⊢ False",
"ppTerm": "?m.193",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nh : S.multiplicity = 1\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 249,
"column": 6
} | {
"line": 249,
"column": 17
} | {
"line": 249,
"column": 18
} | [
{
"pp": "case pos\nK : Type u_4\ninst✝¹ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝ : AdmissibleAbsValues K\nh✝¹ : IsEmpty ι'\nh✝ : archAbsVal.card = 0\n⊢ 1 * ∏ᶠ (v : ↑nonarchAbsVal), ⨆ j, 1 ≤ 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"... | [
"case pos\nK : Type u_4\ninst✝¹ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝ : AdmissibleAbsValues K\nh✝¹ : IsEmpty ι'\nh✝ : archAbsVal.card = 0\n⊢ ∏ᶠ (v : ↑nonarchAbsVal), 0 ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.Projectivization | {
"line": 34,
"column": 4
} | {
"line": 34,
"column": 19
} | {
"line": 34,
"column": 20
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\na b : { v // v ≠ 0 }\nt : K\nh : t = 0\n⊢ ↑a ≠ t • ↑b",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"congrArg",
"... | [
"K : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\na b : { v // v ≠ 0 }\nt : K\nh : t = 0\n⊢ ¬↑a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 19
} | {
"line": 229,
"column": 2
} | [
{
"pp": "s : ℝ\nhs : 1 < s\n⊢ Tendsto\n (fun N ↦\n 1 / (s - 1) * (1 - 1 / (↑N + 1) ^ (s - 1)) - 1 / s * (∑ n ∈ Finset.range N, 1 / (↑n + 1) ^ s - ↑N / (↑N + 1) ^ s))\n atTop (𝓝 (1 / (s - 1) - 1 / s * ∑' (n : ℕ), 1 / (↑n + 1) ^ s))",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [... | [
"case hf\ns : ℝ\nhs : 1 < s\n⊢ Tendsto (fun x ↦ 1 / (s - 1) * (1 - 1 / (↑x + 1) ^ (s - 1))) atTop (𝓝 (1 / (s - 1)))",
"case hg\ns : ℝ\nhs : 1 < s\n⊢ Tendsto (fun x ↦ 1 / s * (∑ n ∈ Finset.range x, 1 / (↑n + 1) ^ s - ↑x / (↑x + 1) ^ s)) atTop\n (𝓝 (1 / s * ∑' (n : ℕ), 1 / (↑n + 1) ^ s))"
] | apply Tendsto.sub | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.RootsOfUnity.Lemmas | {
"line": 39,
"column": 2
} | {
"line": 40,
"column": 83
} | {
"line": 40,
"column": 84
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nμ : R\nhμ : IsPrimitiveRoot μ (n + 1)\nthis : eval 1 (∏ k ∈ range n, (X - C (μ ^ (k + 1) * 1))) = eval 1 (∑ i ∈ range (n + 1), X ^ i)\n⊢ ∏ k ∈ range n, (1 - μ ^ (k + 1)) = ↑n + 1",
"ppTerm": "?m.186",
"assigned": false,
"usedCons... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nμ : R\nhμ : IsPrimitiveRoot μ (n + 1)\nthis : eval 1 (∏ k ∈ range n, (X - C (μ ^ (k + 1) * 1))) = eval 1 (∑ i ∈ range (n + 1), X ^ i)\n⊢ ∏ k ∈ range n, (1 - μ ^ (k + 1)) = ↑n + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 304,
"column": 6
} | {
"line": 304,
"column": 17
} | {
"line": 304,
"column": 18
} | [
{
"pp": "case inl.inr\nK : Type u_4\ninst✝³ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝² : AdmissibleAbsValues K\ninst✝¹ : Finite ι'\ninst✝ : Finite ι\nN : ℕ\np : ι' → MvPolynomial ι K\nhp : ∀ (i : ι'), (p i).IsHomogeneous N\nh : (fun j ↦ constantCoeff (p j)) ≠ 0\n⊢ (mulHeight fun j ↦ (eval 0) (p j)) ≤ max (m... | [
"case inl.inr\nK : Type u_4\ninst✝³ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝² : AdmissibleAbsValues K\ninst✝¹ : Finite ι'\ninst✝ : Finite ι\nN : ℕ\np : ι' → MvPolynomial ι K\nhp : ∀ (i : ι'), (p i).IsHomogeneous N\nh : (fun j ↦ constantCoeff (p j)) ≠ 0\n⊢ (mulHeight fun j ↦ constantCoeff (p j)) ≤ mulHeightBoun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 36
} | {
"line": 307,
"column": 37
} | [
{
"pp": "case inr.inl\nK : Type u_4\ninst✝³ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝² : AdmissibleAbsValues K\ninst✝¹ : Finite ι'\ninst✝ : Finite ι\nN : ℕ\np : ι' → MvPolynomial ι K\nhp : ∀ (i : ι'), (p i).IsHomogeneous N\nx : ι → K\nhx : x ≠ 0\nh₀ : (fun j ↦ (eval x) (p j)) = 0\n⊢ (mulHeight fun j ↦ (eval... | [
"case inr.inl\nK : Type u_4\ninst✝³ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝² : AdmissibleAbsValues K\ninst✝¹ : Finite ι'\ninst✝ : Finite ι\nN : ℕ\np : ι' → MvPolynomial ι K\nhp : ∀ (i : ι'), (p i).IsHomogeneous N\nx : ι → K\nhx : x ≠ 0\nh₀ : (fun j ↦ (eval x) (p j)) = 0\n⊢ 1 ≤ mulHeight x ^ N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.Lemmas | {
"line": 64,
"column": 2
} | {
"line": 71,
"column": 34
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nk n : ℕ\nμ : R\nm : ℕ\nhn : k < m + k + 1\nhμ : IsPrimitiveRoot μ (m + k + 1)\nhdvd : ∀ (k : ℕ), ∃ z ∈ ℤ[μ], μ ^ k - 1 = z * (μ - 1)\nZ : ℕ → R := fun k ↦ Classical.choose ⋯\nZdef : ∀ (k : ℕ), Z k ∈ ℤ[μ] ∧ μ ^ k - 1 = Z k * (μ - 1)\n... | [
"case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nk n : ℕ\nμ : R\nm : ℕ\nhn : k < m + k + 1\nhμ : IsPrimitiveRoot μ (m + k + 1)\nhdvd : ∀ (k : ℕ), ∃ z ∈ ℤ[μ], μ ^ k - 1 = z * (μ - 1)\nZ : ℕ → R := fun k ↦ Classical.choose ⋯\nZdef : ∀ (k : ℕ), Z k ∈ ℤ[μ] ∧ μ ^ k - 1 = Z k * (μ - 1)\n⊢ ↑(m + k + ... | · apply Subalgebra.mul_mem
· apply Subalgebra.mul_mem
· exact Subalgebra.pow_mem _ (Subalgebra.neg_mem _ <| Subalgebra.one_mem _) _
· exact Subalgebra.prod_mem _ fun _ _ ↦ (Zdef _).1
· refine Subalgebra.prod_mem _ fun _ _ ↦ ?_
apply Subalgebra.sub_mem
· exact Subalgebra.pow_mem _ (self_m... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 324,
"column": 50
} | {
"line": 324,
"column": 61
} | {
"line": 324,
"column": 62
} | [
{
"pp": "this : Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 γ)\ns : ℝ\nhs : s ∈ Ioi 1\n⊢ 1 < (↑s).re",
"ppTerm": "?m.130",
"assigned": true,
"usedConstants": [
"Real",
"Real.instLT",
"id",
"Complex.ofReal",
"Complex.re",
"Real.instOn... | [
"this : Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 γ)\ns : ℝ\nhs : s ∈ Ioi 1\n⊢ 1 < s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 332,
"column": 4
} | {
"line": 332,
"column": 27
} | {
"line": 332,
"column": 28
} | [
{
"pp": "aux2 : Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 (1 - termTSum 1))\nthis : γ = 1 - termTSum 1\n⊢ Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 γ)",
"ppTerm": "?m.266",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"R... | [
"aux2 : Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 (1 - termTSum 1))\nthis : γ = 1 - termTSum 1\n⊢ Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 (1 - termTSum 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 367,
"column": 6
} | {
"line": 367,
"column": 23
} | {
"line": 368,
"column": 6
} | [
{
"pp": "case refine_2.refine_2\nf : ℂ → ℂ := fun s ↦ riemannZeta s - 1 / (s - 1)\n⊢ Tendsto (fun x ↦ f x * (x - 1) - f 1 * (x - 1)) (𝓝[≠] 1) (𝓝 (1 - 1 - f 1 * (1 - 1)))",
"ppTerm": "?refine_2.refine_2",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"NonUn... | [
"case refine_2.refine_2.hf\nf : ℂ → ℂ := ⋯\n⊢ Tendsto (fun x ↦ f x * (x - 1)) (𝓝[≠] 1) (𝓝 (1 - 1))",
"case refine_2.refine_2.hg\nf : ℂ → ℂ := ⋯\n⊢ Tendsto (fun x ↦ f 1 * (x - 1)) (𝓝[≠] 1) (𝓝 (f 1 * (1 - 1)))"
] | apply Tendsto.sub | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 55
} | {
"line": 378,
"column": 5
} | [
{
"pp": "F : Type u_1\ninst✝² : Norm F\ninst✝¹ : One F\ninst✝ : NormOneClass F\n⊢ (fun s ↦ riemannZeta s - 1 / (s - 1)) =O[𝓝 1] fun x ↦ 1",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"F : Type u_1\ninst✝² : Norm F\ninst✝¹ : One F\ninst✝ : NormOneClass F\n⊢ (fun s ↦ riemannZeta s - 1 / (s - 1)) =O[𝓝 1] fun x ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 94
} | {
"line": 111,
"column": 95
} | [
{
"pp": "case hc\nF : Type u_1\nR : Type u_2\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : CommRing R\nx : F\nhx : x ≠ 0 ∧ x ≠ 1\n⊢ IsUnit (x * (1 - x))",
"ppTerm": "?hc",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.toMonoidWithZer... | [
"case hc\nF : Type u_1\nR : Type u_2\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : CommRing R\nx : F\nhx : x ≠ 0 ∧ x ≠ 1\n⊢ ¬x = 0 ∧ ¬x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 63,
"column": 2
} | {
"line": 64,
"column": 54
} | {
"line": 64,
"column": 55
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : DecidableEq (AbsoluteValue K ℝ)\nv : InfinitePlace K\nthis : DecidableEq (InfinitePlace K)\n⊢ Multiset.count (↑v) (multisetInfinitePlace K) = v.mult",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Multiset.sum"... | [
"K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : DecidableEq (AbsoluteValue K ℝ)\nv : InfinitePlace K\nthis : DecidableEq (InfinitePlace K)\n⊢ (∑ x, if x = v then x.mult else 0) = v.mult"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 147,
"column": 2
} | {
"line": 148,
"column": 9
} | {
"line": 148,
"column": 10
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : Inhabited (InfinitePlace K)\n⊢ 0 < totalWeight K",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Finset.univ",
"Multiset.map",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : Inhabited (InfinitePlace K)\n⊢ 0 < ∑ x, x.mult"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 482,
"column": 4
} | {
"line": 482,
"column": 15
} | {
"line": 482,
"column": 16
} | [
{
"pp": "case inr\ns : ℂ\nhs : s ≠ 1\nhg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ\nhgz : ∀ (z : ℂ), 1 < z.re → (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z)) = riemannZeta z\nheq : EqOn (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) riemannZeta {1}ᶜ\n... | [
"case inr\ns : ℂ\nhs : s ≠ 1\nhg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ\nhgz : ∀ (z : ℂ), 1 < z.re → (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z)) = riemannZeta z\nheq : EqOn (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) riemannZeta {1}ᶜ\n⊢ riemannZet... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 22
} | {
"line": 453,
"column": 23
} | [
{
"pp": "K : Type u_6\ninst✝³ : Field K\nι : Type u_7\nι' : Type u_8\ninst✝² : Fintype ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Finite ι\nM N : ℕ\nq : ι × ι' → MvPolynomial ι K\nhq : ∀ (a : ι × ι'), (q a).IsHomogeneous M\np : ι' → MvPolynomial ι K\nx : ι → K\nh : ∀ (k : ι), ∑ j, (eval x) (q (k, j)) * (eval x... | [
"K : Type u_6\ninst✝³ : Field K\nι : Type u_7\nι' : Type u_8\ninst✝² : Fintype ι'\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Finite ι\nM N : ℕ\nq : ι × ι' → MvPolynomial ι K\nhq : ∀ (a : ι × ι'), (q a).IsHomogeneous M\np : ι' → MvPolynomial ι K\nx : ι → K\nh : ∀ (k : ι), ∑ j, (eval x) (q (k, j)) * (eval x) (p j) = x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 13
} | {
"line": 566,
"column": 14
} | [
{
"pp": "⊢ Tendsto riemannZeta₁ (𝓝 1) (𝓝 1)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ Tendsto riemannZeta₁ (𝓝 1) (𝓝 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 611,
"column": 4
} | {
"line": 611,
"column": 15
} | {
"line": 611,
"column": 16
} | [
{
"pp": "this : DifferentiableAt ℝ (fun s ↦ Real.log (riemannZeta₁ ↑s).re) 1\n⊢ (fun s ↦ Real.log (riemannZeta₁ ↑s).re) =O[𝓝 1] fun x ↦ x - 1",
"ppTerm": "?m.101",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"this : DifferentiableAt ℝ (fun s ↦ Real.log (riemannZeta₁ ↑s).re) 1\n⊢ (fun s ↦ Real.log (riemannZeta₁ ↑s).re) =O[𝓝 1] fun x ↦ x - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 635,
"column": 4
} | {
"line": 635,
"column": 15
} | {
"line": 635,
"column": 16
} | [
{
"pp": "this : DifferentiableAt ℂ (fun s ↦ deriv riemannZeta₁ s / riemannZeta₁ s) 1\n⊢ (fun s ↦ deriv riemannZeta₁ s / riemannZeta₁ s - ↑γ) =O[𝓝 1] fun x ↦ x - 1",
"ppTerm": "?m.151",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"this : DifferentiableAt ℂ (fun s ↦ deriv riemannZeta₁ s / riemannZeta₁ s) 1\n⊢ (fun s ↦ deriv riemannZeta₁ s / riemannZeta₁ s - ↑γ) =O[𝓝 1] fun x ↦ x - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 47
} | {
"line": 255,
"column": 48
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 662,
"column": 4
} | {
"line": 662,
"column": 25
} | {
"line": 662,
"column": 26
} | [
{
"pp": "this : (fun s ↦ (riemannZeta₁ s)⁻¹ - 1) =O[𝓝 1] fun x ↦ x - 1\n⊢ (fun s ↦ (s - 1) * ((riemannZeta₁ s)⁻¹ - 1)) =O[𝓝 1] fun s ↦ (s - 1) ^ 2",
"ppTerm": "?m.187",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"riemannZeta₁",
"HM... | [
"this : (fun s ↦ (riemannZeta₁ s)⁻¹ - 1) =O[𝓝 1] fun x ↦ x - 1\n⊢ (fun s ↦ (s - 1) * ((riemannZeta₁ s)⁻¹ - 1)) =O[𝓝 1] fun s ↦ (s - 1) * (s - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 13
} | {
"line": 663,
"column": 14
} | [
{
"pp": "⊢ (fun s ↦ (riemannZeta₁ s)⁻¹ - 1) =O[𝓝 1] fun x ↦ x - 1",
"ppTerm": "?m.186",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ (fun s ↦ (riemannZeta₁ s)⁻¹ - 1) =O[𝓝 1] fun x ↦ x - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 669,
"column": 4
} | {
"line": 669,
"column": 25
} | {
"line": 669,
"column": 26
} | [
{
"pp": "this : (fun x ↦ x - 1) =o[𝓝 1] fun x ↦ 1\n⊢ (fun s ↦ (s - 1) ^ 2) =o[𝓝[≠] 1] fun x ↦ x - 1",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"HMul.hMul",
"pow_two",
"Monoid.toMulOneClass",
"congrA... | [
"this : (fun x ↦ x - 1) =o[𝓝 1] fun x ↦ 1\n⊢ (fun s ↦ (s - 1) * (s - 1)) =o[𝓝[≠] 1] fun x ↦ x - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 117,
"column": 56
} | {
"line": 117,
"column": 67
} | {
"line": 117,
"column": 68
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRoot S f\np : R[X]\n⊢ h.map p = 0 ↔ f ∣ p",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRoot S f\np : R[X]\n⊢ h.map p = 0 ↔ f ∣ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 176,
"column": 74
} | {
"line": 177,
"column": 92
} | {
"line": 179,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRoot S f\na : AdjoinRoot f\n⊢ h.adjoinRootAlgEquiv a = h.map ⋯.choose",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Exists.choose_spec",
"AdjoinR... | [] | by
rw (occs := [1]) [← (AdjoinRoot.mk_surjective a).choose_spec, adjoinRootAlgEquiv_apply_mk] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Height.NumberField | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 78
} | {
"line": 271,
"column": 78
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nn : ℕ\nhn : ¬n = 0\na : 𝓞 K\nHw : 0 < absNorm (span {↑n})\n⊢ 1 ≤ ↑(absNorm (span {↑n})) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i)",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.t... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nn : ℕ\nhn : ¬n = 0\na : 𝓞 K\nHw : 0 < absNorm (span {↑n})\n⊢ ↑(absNorm (span (Set.range ![a, ↑n]))) * ∏ᶠ (v : FinitePlace K), ⨆ i, v ↑(![a, ↑n] i) ≤\n ↑(absNorm (span {↑n})) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i)"
] | ← absNorm_mul_finprod_finitePlace_eq_one (show ![a, n] ≠ 0 by simp [hn]) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 42
} | {
"line": 407,
"column": 43
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nn : ℕ\na✝ : Nontrivial R\nhdeg : f.degree ≤ ↑n\n⊢ f.natDegree ≤ n",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
... | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nn : ℕ\na✝ : Nontrivial R\nhdeg : f.degree ≤ ↑n\n⊢ f.degree ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 13
} | {
"line": 411,
"column": 14
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nhdeg : 1 < f.natDegree\n⊢ h.modByMonicHom h.root = X",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nhdeg : 1 < f.natDegree\n⊢ h.modByMonicHom h.root = X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 299,
"column": 4
} | {
"line": 299,
"column": 25
} | {
"line": 299,
"column": 26
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nn : ℕ\nhn : n ≠ 0\na : 𝓞 K\nha₁ : ↑n * x = ↑a\nha₂ : Ideal.absNorm (Ideal.span {↑n, a}) = n ^ (totalWeight K - 1)\nhv : ∀ (i : Fin 2), ↑(![a, ↑n] i) = ![↑a, ↑n] i\n⊢ ↑n ^ (totalWeight K - 1) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i) = ... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nn : ℕ\nhn : n ≠ 0\na : 𝓞 K\nha₁ : ↑n * x = ↑a\nha₂ : Ideal.absNorm (Ideal.span {↑n, a}) = n ^ (totalWeight K - 1)\nhv : ∀ (i : Fin 2), ↑(![a, ↑n] i) = ![↑a, ↑n] i\n⊢ ↑n ^ (totalWeight K - 1) * ∏ᶠ (v : FinitePlace K), ⨆ i, v (![↑a, ↑n] i) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 297,
"column": 18
} | {
"line": 297,
"column": 35
} | {
"line": 297,
"column": 35
} | [
{
"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... | jacobiSum_one_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 602,
"column": 4
} | {
"line": 602,
"column": 42
} | {
"line": 602,
"column": 43
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nf : R[X]\nh : IsAdjoinRoot S f\ninst✝³ : IsDomain R\ninst✝² : IsDomain S\ninst✝¹ : IsTorsionFree R S\ninst✝ : IsIntegrallyClosed R\nα : S\nhα : IsIntegral R α\nhα₂ : R[α] = ⊤\nx✝ : R[X]\n⊢ x✝ ∈ RingHom.ker (aeval α)... | [
"R : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nf : R[X]\nh : IsAdjoinRoot S f\ninst✝³ : IsDomain R\ninst✝² : IsDomain S\ninst✝¹ : IsTorsionFree R S\ninst✝ : IsIntegrallyClosed R\nα : S\nhα : IsIntegral R α\nhα₂ : R[α] = ⊤\nx✝ : R[X]\n⊢ (aeval α) x✝ = 0 ↔ minpoly R α ∣ x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 474,
"column": 4
} | {
"line": 474,
"column": 23
} | {
"line": 474,
"column": 23
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nhx : Finset.univ.gcd x = 1\nhx₀ : Int.cast ∘ x ≠ 0\n⊢ (∏ v, (⨆ i, v ((Int.cast ∘ x) i)) ^ v.mult) * ∏ᶠ (v : FinitePlace ℚ), ⨆ i, v ((Int.cast ∘ x) i) = ⨆ i, ↑|x i|",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nhx : Finset.univ.gcd x = 1\nhx₀ : Int.cast ∘ x ≠ 0\n⊢ (∏ x_1, (⨆ i, ↑|(Int.cast ∘ x) i|) ^ x_1.mult) * ∏ᶠ (v : FinitePlace ℚ), ⨆ i, v ((Int.cast ∘ x) i) = ⨆ i, ↑|x i|"
] | infinitePlace_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 499,
"column": 4
} | {
"line": 499,
"column": 84
} | {
"line": 500,
"column": 6
} | [
{
"pp": "q : ℚ\n⊢ Finset.univ.gcd ![q.num, ↑q.den] = 1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"Rat.num",
"Finset.univ",
"congrArg",
"Int.instStrongNormalizedGCDMonoid",
"Finset",
"instNormalizedGCDMonoidOfStrongN... | [
"q : ℚ\n⊢ q.num.gcd ↑q.den = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 621,
"column": 40
} | {
"line": 621,
"column": 62
} | {
"line": 621,
"column": 62
} | [
{
"pp": "K : Type u_4\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nC₁ : ℝ\nhC₁ : ∀ (a b c d : K), logHeight ![a * c, a * d + b * c, b * d] ≤ C₁ + logHeight ![a, b] + logHeight ![c, d]\nC₂ : ℝ\nhC₂ :\n ∀ {a b c d : K},\n ![a, b] ≠ 0 → ![c, d] ≠ 0 → C₂ + logHeight ![a, b] + logHeight ![c, d] ≤ logHeight ... | [
"K : Type u_4\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nC₁ : ℝ\nhC₁ : ∀ (a b c d : K), logHeight ![a * c, a * d + b * c, b * d] ≤ C₁ + logHeight ![a, b] + logHeight ![c, d]\nC₂ : ℝ\na✝ b✝ c✝ d✝ : K\nhC₂ : C₂ + logHeight ![a✝, b✝] + logHeight ![c✝, d✝] ≤ logHeight ![a✝ * c✝, a✝ * d✝ + b✝ * c✝, b✝ * d✝]\nhab ... | specialize hC₂ hab hcd | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 45
} | {
"line": 144,
"column": 46
} | [
{
"pp": "case refine_3.refine_2\nM N : ℕ\ninst✝¹ : NeZero M\ninst✝ : NeZero N\nhMN : M ∣ N\nχ : DirichletCharacter ℂ M\ns : ℂ\nhs : s ≠ 1\nhpc : IsPreconnected {1}ᶜ\nhne : 2 ∈ {1}ᶜ\nt : ℂ\nht : 1 < t.re\n⊢ LFunction ((changeLevel hMN) χ) t = (fun s ↦ LFunction χ s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s)))... | [
"case refine_3.refine_2\nM N : ℕ\ninst✝¹ : NeZero M\ninst✝ : NeZero N\nhMN : M ∣ N\nχ : DirichletCharacter ℂ M\ns : ℂ\nhs : s ≠ 1\nhpc : IsPreconnected {1}ᶜ\nhne : 2 ∈ {1}ᶜ\nt : ℂ\nht : 1 < t.re\n⊢ LSeries (fun x ↦ ((changeLevel hMN) χ) ↑x) t = LSeries (fun x ↦ χ ↑x) t * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-t))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 9
} | {
"line": 138,
"column": 10
} | [
{
"pp": "k : ℕ\nx : ℝ\n⊢ bernoulliFun k (1 - x) = (-1) ^ k * bernoulliFun k x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"bernoulliFun._proof_1",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.bernoulli",
... | [
"k : ℕ\nx : ℝ\n⊢ (Polynomial.aeval (1 - x)) (Polynomial.bernoulli k) = (-1) ^ k * (Polynomial.aeval x) (Polynomial.bernoulli k)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 41
} | {
"line": 196,
"column": 42
} | [
{
"pp": "case neg\nN : 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}\nhUo : IsOpen U\nf : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)\ng : ℂ → ℂ := expZeta (toAddCircle j)\nhU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0\nhf : Anal... | [
"case neg\nN : 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}\nhUo : IsOpen U\nf : ℂ → ℂ := LFunction fun k ↦ 𝕖 (j * k)\ng : ℂ → ℂ := expZeta (toAddCircle j)\nhU : ∀ {u : ℂ}, u ∈ U ↔ u ≠ 1 ∨ j ≠ 0\nhf : AnalyticOnNhd ℂ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 24
} | {
"line": 189,
"column": 25
} | [
{
"pp": "case succ\nm : ℕ\nm0 : m ≠ 0\nx : ℝ\nm0' : ↑m ≠ 0\nf : ℕ → ℝ → ℝ := fun k x ↦ bernoulliFun k (↑m * x) - ↑m ^ k / ↑m * ∑ i ∈ Finset.range m, bernoulliFun k (x + ↑i / ↑m)\nk : ℕ\nh : ∀ (x : ℝ), f k x = 0\nd : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x\nc : ℝ\nfc : ∀ (x : ℝ), f (k + 1) x = c\ni : c = 0\n⊢ ∀ (x... | [
"case succ\nm : ℕ\nm0 : m ≠ 0\nx : ℝ\nm0' : ↑m ≠ 0\nf : ℕ → ℝ → ℝ := fun k x ↦ bernoulliFun k (↑m * x) - ↑m ^ k / ↑m * ∑ i ∈ Finset.range m, bernoulliFun k (x + ↑i / ↑m)\nk : ℕ\nh : ∀ (x : ℝ), f k x = 0\nd : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x\nc : ℝ\nfc : ∀ (x : ℝ), f (k + 1) x = c\ni : c = 0\n⊢ ∀ (x : ℝ), f (k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 206,
"column": 58
} | {
"line": 206,
"column": 68
} | {
"line": 206,
"column": 69
} | [
{
"pp": "case neg\nk : ℕ\nk1 : ¬k = 1\nm : bernoulliFun k 1 = 2 ^ k / 2 * (0 + bernoulliFun k (2⁻¹ + 0 / 2) + bernoulliFun k (2⁻¹ + 1 / 2))\n⊢ bernoulliFun k 2⁻¹ = (2 / 2 ^ k - 1) * ↑(bernoulli k)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
... | [
"case neg\nk : ℕ\nk1 : ¬k = 1\nm : bernoulliFun k 1 = 2 ^ k / 2 * (0 + bernoulliFun k (1 / 2 + 0 / 2) + bernoulliFun k (1 / 2 + 1 / 2))\n⊢ bernoulliFun k 2⁻¹ = (2 / 2 ^ k - 1) * ↑(bernoulli k)"
] | ← one_div, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 13
} | {
"line": 241,
"column": 14
} | [
{
"pp": "n : ℤ\nhn : n ≠ 0\n⊢ bernoulliFourierCoeff 0 n = 0",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℤ\nhn : n ≠ 0\n⊢ bernoulliFourierCoeff 0 n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 27
} | {
"line": 345,
"column": 28
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\np : ℕ\nhp : p ∈ n.primeFactors\n⊢ 1 - (↑p)⁻¹ ≠ 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneCl... | [
"n : ℕ\ninst✝ : NeZero n\np : ℕ\nhp : p ∈ n.primeFactors\n⊢ ¬p = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 13
} | {
"line": 355,
"column": 14
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ ContinuousWithinAt (LFunctionTrivChar₁ n) {1}ᶜ 1",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"ContinuousWithinAt",
"HMul.hMul",
"Complex.instNormedAddCommGroup",
"C... | [
"n : ℕ\ninst✝ : NeZero n\n⊢ Tendsto (fun s ↦ (s - 1) * LFunctionTrivChar n s) (𝓝[≠] 1) (𝓝 (∏ p ∈ n.primeFactors, (1 - (↑p)⁻¹)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 49
} | {
"line": 42,
"column": 50
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ LSeriesSummable (f + g) s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"id",
"instHAdd",
"Pi.instAdd",
"HAdd.hAdd",
"Nat",
"Complex.instAdd",
"Complex",
"LSe... | [
"f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ Summable (term (f + g) s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 33
} | {
"line": 47,
"column": 34
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ LSeries (f + g) s = LSeries f s + LSeries g s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"congrArg",
"SummationFilter",
"Co... | [
"f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ ∑' (n : ℕ), (term f s n + term g s n) = ∑' (n : ℕ), term f s n + ∑' (n : ℕ), term g s n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 49
} | {
"line": 96,
"column": 50
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ LSeriesSummable (f - g) s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"HSub.hSub",
"id",
"instHSub",
"Nat",
"Pi.instSub",
"Complex.instSub",
"Complex",
"LSe... | [
"f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ Summable (term (f - g) s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 33
} | {
"line": 101,
"column": 34
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ LSeries (f - g) s = LSeries f s - LSeries g s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"congrArg",
"SummationFilter",
"Co... | [
"f g : ℕ → ℂ\ns : ℂ\nhf : LSeriesSummable f s\nhg : LSeriesSummable g s\n⊢ ∑' (n : ℕ), (term f s n - term g s n) = ∑' (n : ℕ), term f s n - ∑' (n : ℕ), term g s n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 18
} | {
"line": 125,
"column": 19
} | [
{
"pp": "f : ℕ → ℂ\nc s : ℂ\nhc : c ≠ 0\nhf : LSeriesSummable (c • f) s\n⊢ LSeriesSummable f s",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : ℕ → ℂ\nc s : ℂ\nhc : c ≠ 0\nhf : LSeriesSummable (c • f) s\n⊢ LSeriesSummable f s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 77
} | {
"line": 156,
"column": 78
} | [
{
"pp": "ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\na : ι → ℂ\nhf : ∀ i ∈ S, LSeriesHasSum (f i) s (a i)\n⊢ LSeriesHasSum (∑ i ∈ S, f i) s (∑ i ∈ S, a i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LSeries.term_sum",
"NormedCommRing.toSeminormedCommR... | [
"ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\na : ι → ℂ\nhf : ∀ i ∈ S, LSeriesHasSum (f i) s (a i)\n⊢ HasSum (fun a ↦ ∑ c ∈ S, term (f c) s a) (∑ i ∈ S, a i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 49
} | {
"line": 160,
"column": 50
} | [
{
"pp": "ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\nhf : ∀ i ∈ S, LSeriesSummable (f i) s\n⊢ LSeriesSummable (∑ i ∈ S, f i) s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Pi.addCommMonoid",
"id",
"Nat",
"Complex",
"Complex.instAddCommMonoid",
... | [
"ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\nhf : ∀ i ∈ S, LSeriesSummable (f i) s\n⊢ Summable (term (∑ i ∈ S, f i) s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 33
} | {
"line": 165,
"column": 34
} | [
{
"pp": "ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\nhf : ∀ i ∈ S, LSeriesSummable (f i) s\n⊢ LSeries (∑ i ∈ S, f i) s = ∑ i ∈ S, LSeries (f i) s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Pi.addCommMonoid",
... | [
"ι : Type u_1\nf : ι → ℕ → ℂ\nS : Finset ι\ns : ℂ\nhf : ∀ i ∈ S, LSeriesSummable (f i) s\n⊢ ∑' (n : ℕ), ∑ i ∈ S, term (f i) s n = ∑ x ∈ S, ∑' (n : ℕ), term (f x) s n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 32
} | {
"line": 72,
"column": 33
} | [
{
"pp": "f : ℕ → ℂ\nn : ℕ\nh : ∀ m ≤ n, f m = 0\nha : abscissaOfAbsConv f < ⊤\ny : ℝ\nhay : abscissaOfAbsConv f < ↑y\nhyt : ↑y < ⊤\nF : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x\nhF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0\nhF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (... | [
"f : ℕ → ℂ\nn : ℕ\nh : ∀ m ≤ n, f m = 0\nha : abscissaOfAbsConv f < ⊤\ny : ℝ\nhay : abscissaOfAbsConv f < ↑y\nhyt : ↑y < ⊤\nF : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x\nhF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0\nhF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 439,
"column": 72
} | {
"line": 442,
"column": 70
} | {
"line": 444,
"column": 2
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Odd Φ\ns : ℂ\nhs : 1 < s.re\nthis : ∑ x, Φ x * sinZeta (toAddCircle x) s = I * LFunction (𝓕 Φ) s\n⊢ ∑ x, Φ x * completedSinZeta (toAddCircle x) s = I * completedLFunction (𝓕 Φ) s",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants"... | [] | by
have hs' : 0 < re (s + 1) := by simp only [add_re, one_re]; linarith
simpa only [sinZeta, ← mul_div_assoc, ← sum_div, div_left_inj' (Gammaℝ_ne_zero_of_re_pos hs'),
LFunction_eq_completed_div_gammaFactor_odd (dft_odd_iff.mpr hΦ)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 13
} | {
"line": 183,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : x ∈ Icc 0 1\nk : ℕ\nhk : k.succ ≠ 0\n⊢ hurwitzZetaEven (↑x) (-(2 * ↑k.succ)) = 0",
"ppTerm": "?m.111",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
... | [
"x : ℝ\nhx : x ∈ Icc 0 1\nk : ℕ\nhk : k.succ ≠ 0\n⊢ hurwitzZetaEven (↑x) (-(2 * (↑k + 1))) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 86,
"column": 2
} | {
"line": 87,
"column": 7
} | {
"line": 87,
"column": 8
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nn : ℕ\nhn : n ≠ 0\n⊢ ‖(toArithmeticFunction fun x ↦ χ ↑x) n‖ ≤ 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Nat.instMulZeroClass",
"Real.instLE",
"Real",
"toAr... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nn : ℕ\nhn : n ≠ 0\n⊢ ‖χ ↑n‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 50
} | {
"line": 184,
"column": 51
} | [
{
"pp": "case inr.refine_2.hf\nN : ℕ\ninst✝ : NeZero N\nB : BadChar N\nG : ℂ → ℂ := ⋯\nH : ℂ → ℂ := ⋯\nthis : B.F = G * H\n⊢ ContinuousAt G 1",
"ppTerm": "?inr.refine_2.hf",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"HMul.hMul",
"ri... | [
"case inr.refine_2.hf\nN : ℕ\ninst✝ : NeZero N\nB : BadChar N\nG : ℂ → ℂ := Function.update (fun s ↦ (s - 1) * riemannZeta s) 1 1\nH : ℂ → ℂ := Function.update (fun s ↦ (LFunction B.χ s - LFunction B.χ 1) / (s - 1)) 1 (deriv (LFunction B.χ) 1)\nthis : B.F = G * H\n⊢ Tendsto (fun s ↦ (s - 1) * riemannZeta s) (𝓝[≠] ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 502,
"column": 4
} | {
"line": 502,
"column": 46
} | {
"line": 502,
"column": 47
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Even Φ\ns : ℂ\nhs₀ : s ≠ 0 ∨ ∑ j, Φ j = 0\nhs₁ : s ≠ 1 ∨ Φ 0 = 0\nF : ℂ → ℂ := fun t ↦ completedLFunction Φ (1 - t)\nG : ℂ → ℂ := fun t ↦ ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t\nU : Set ℂ := {t | (t ≠ 0 ∨ ∑ j, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}\nhsU... | [
"N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Even Φ\ns : ℂ\nhs₀ : s ≠ 0 ∨ ∑ j, Φ j = 0\nhs₁ : s ≠ 1 ∨ Φ 0 = 0\nF : ℂ → ℂ := fun t ↦ completedLFunction Φ (1 - t)\nG : ℂ → ℂ := fun t ↦ ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t\nU : Set ℂ := {t | (t ≠ 0 ∨ ∑ j, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}\nhsU : s ∈ U\nh2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 103,
"column": 28
} | {
"line": 103,
"column": 49
} | {
"line": 104,
"column": 6
} | [
{
"pp": "f : ℕ → ℂ\nn : ℕ\nh : ∀ m ≤ n, f m = 0\nha : abscissaOfAbsConv f < ⊤\ny : ℝ\nhay : abscissaOfAbsConv f < ↑y\nhyt : ↑y < ⊤\nF : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x\nhF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0\nhF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (... | [
"f : ℕ → ℂ\nn : ℕ\nh : ∀ m ≤ n, f m = 0\nha : abscissaOfAbsConv f < ⊤\ny : ℝ\nhay : abscissaOfAbsConv f < ↑y\nhyt : ↑y < ⊤\nF : ℝ → ℕ → ℂ := fun x ↦ {m | n + 1 < m}.indicator fun m ↦ f m / (↑m / (↑n + 1)) ^ ↑x\nhF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0\nhF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x... | rw [← mul_zero (f k)] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 276,
"column": 2
} | {
"line": 277,
"column": 9
} | {
"line": 277,
"column": 10
} | [
{
"pp": "q : ℕ\na : ZMod q\ninst✝ : NeZero q\nha : IsUnit a\nn : ℕ\n⊢ ↑(residueClass a n) = ((↑q.totient)⁻¹ • ∑ χ, χ a⁻¹ • fun n ↦ χ ↑n * ↑(Λ n)) n",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"DirichletCharacter.fintype",
"Eq.mpr",
... | [
"q : ℕ\na : ZMod q\ninst✝ : NeZero q\nha : IsUnit a\nn : ℕ\n⊢ ↑(residueClass a n) = (↑q.totient)⁻¹ * ∑ x, x a⁻¹ * x ↑n * ↑(Λ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 270,
"column": 4
} | {
"line": 271,
"column": 11
} | {
"line": 271,
"column": 12
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\np : Nat.Primes\n⊢ ‖χ ↑↑p * ↑↑p ^ (-s)‖ ≤ ↑↑p ^ (-s).re",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Nat.Prime",
"HMu... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\np : Nat.Primes\n⊢ ‖χ ↑↑p‖ * ↑↑p ^ (-s).re ≤ ↑↑p ^ (-s).re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 321,
"column": 29
} | {
"line": 321,
"column": 93
} | {
"line": 321,
"column": 94
} | [
{
"pp": "q : ℕ\na : ZMod q\ninst✝ : NeZero q\nχ : DirichletCharacter ℂ q\nhχ : χ ∈ {1}ᶜ\n⊢ χ ≠ 1",
"ppTerm": "?m.168",
"assigned": true,
"usedConstants": [
"ZMod.commRing",
"MulChar.hasOne",
"id",
"Ne",
"Field.toSemifield",
"ZMod",
"Semifield.toCommGroupWith... | [
"q : ℕ\na : ZMod q\ninst✝ : NeZero q\nχ : DirichletCharacter ℂ q\nhχ : χ ∈ {1}ᶜ\n⊢ ¬χ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 225,
"column": 2
} | {
"line": 226,
"column": 9
} | {
"line": 226,
"column": 10
} | [
{
"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": false,
"us... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 300,
"column": 2
} | {
"line": 301,
"column": 44
} | {
"line": 302,
"column": 6
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\nx : ℝ\nhx : 0 < x\ny : ℝ\nh₀ : 1 < (1 + ↑x).re\nh₁ : 1 < (1 + ↑x + I * ↑y).re\nh₂ : 1 < (1 + ↑x + 2 * I * ↑y).re\nH₀ : Summable fun p ↦ -log (1 - 1 ↑↑p * ↑↑p ^ (-(1 + ↑x)))\nH₁ : Summable fun p ↦ -log (1 - χ ↑↑p * ↑↑p ^ (-(1 + ↑x + I * ↑y)))\nH₂ : Summable fun p ↦ -lo... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\nx : ℝ\nhx : 0 < x\ny : ℝ\nh₀ : 1 < (1 + ↑x).re\nh₁ : 1 < (1 + ↑x + I * ↑y).re\nh₂ : 1 < (1 + ↑x + 2 * I * ↑y).re\nH₀ : Summable fun p ↦ -log (1 - 1 ↑↑p * ↑↑p ^ (-(1 + ↑x)))\nH₁ : Summable fun p ↦ -log (1 - χ ↑↑p * ↑↑p ^ (-(1 + ↑x + I * ↑y)))\nH₂ : Summable fun p ↦ -log (1 - (χ ^ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 411,
"column": 8
} | {
"line": 411,
"column": 54
} | {
"line": 411,
"column": 55
} | [
{
"pp": "q : ℕ\ninst✝ : NeZero q\na : ZMod q\nha : IsUnit a\nH :\n ∀ {x : ℝ},\n 1 < x → ∑' (n : ℕ), residueClass a n / ↑n ^ x = (LFunctionResidueClassAux a ↑x).re + (↑q.totient)⁻¹ / (x - 1)\nx : ℝ\nhx : x ∈ Set.Icc 1 2\n⊢ ↑x ∈ {s | 1 ≤ s.re}",
"ppTerm": "?m.180",
"assigned": true,
"usedConstants... | [
"q : ℕ\ninst✝ : NeZero q\na : ZMod q\nha : IsUnit a\nH :\n ∀ {x : ℝ},\n 1 < x → ∑' (n : ℕ), residueClass a n / ↑n ^ x = (LFunctionResidueClassAux a ↑x).re + (↑q.totient)⁻¹ / (x - 1)\nx : ℝ\nhx : x ∈ Set.Icc 1 2\n⊢ 1 ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 41,
"column": 2
} | {
"line": 43,
"column": 64
} | {
"line": 45,
"column": 0
} | [
{
"pp": "⊢ riemannZetaZerosᶜ ∈ Filter.codiscreteWithin {1}ᶜ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | refine analyticOn_riemannZeta.preimage_zero_mem_codiscreteWithin (x := 2) ?_ (by simp) ?_
· exact riemannZeta_ne_zero_of_one_le_re Nat.one_le_ofNat
· exact isConnected_compl_singleton_of_one_lt_rank (by simp) 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 41,
"column": 2
} | {
"line": 43,
"column": 64
} | {
"line": 45,
"column": 0
} | [
{
"pp": "⊢ riemannZetaZerosᶜ ∈ Filter.codiscreteWithin {1}ᶜ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | refine analyticOn_riemannZeta.preimage_zero_mem_codiscreteWithin (x := 2) ?_ (by simp) ?_
· exact riemannZeta_ne_zero_of_one_le_re Nat.one_le_ofNat
· exact isConnected_compl_singleton_of_one_lt_rank (by simp) 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 58,
"column": 5
} | {
"line": 58,
"column": 16
} | {
"line": 58,
"column": 17
} | [
{
"pp": "⊢ IsClosed riemannZetaZeros",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ IsClosed riemannZetaZeros"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.ZetaZeros | {
"line": 61,
"column": 5
} | {
"line": 61,
"column": 16
} | {
"line": 61,
"column": 17
} | [
{
"pp": "⊢ IsDiscrete riemannZetaZeros",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ IsDiscrete riemannZetaZeros"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 322,
"column": 8
} | {
"line": 322,
"column": 99
} | {
"line": 323,
"column": 10
} | [
{
"pp": "case convert_2\nN : ℕ\ninst✝ : NeZero N\n⊢ Tendsto (fun w ↦ 1 + w) (𝓝[≠] 0) (𝓝[≠] 1)",
"ppTerm": "?convert_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Compl.compl",
"nhdsWithin",
"PartialOrder.toPreorder",
... | [
"case convert_2\nN : ℕ\ninst✝ : NeZero N\n⊢ 𝓝[≠] 0 ≤ comap (fun w ↦ 1 + w) (𝓝[≠] 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 330,
"column": 2
} | {
"line": 331,
"column": 9
} | {
"line": 331,
"column": 10
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ny : ℝ\nhy : y ≠ 0 ∨ χ ≠ 1\n⊢ 1 + I * ↑y ≠ 1 ∨ χ ≠ 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"False",
"Real",
"HMul.hMul",
"ZMod.commRing",
"MulZeroClass.to... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\ny : ℝ\nhy : y ≠ 0 ∨ χ ≠ 1\n⊢ ¬y = 0 ∨ ¬χ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.Complex | {
"line": 26,
"column": 2
} | {
"line": 26,
"column": 37
} | {
"line": 26,
"column": 38
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\n⊢ ringChar ℂ ≠ ringChar F",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Ne",
"instOfNatNat",
"Field.toSemifield",
"Semifield.toDivisionSemiring",
"ringCh... | [
"F : Type u_1\ninst✝¹ : Field F\ninst✝ : Finite F\n⊢ 0 ≠ ringChar F"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 347,
"column": 2
} | {
"line": 348,
"column": 9
} | {
"line": 348,
"column": 10
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ninst✝ : NeZero N\ny : ℝ\nhy : y ≠ 0 ∨ χ ≠ 1\nh : LFunction χ (1 + I * ↑y) = 0\nthis : HasDerivAt (LFunction χ) (deriv (LFunction χ) (0 + (1 + I * ↑y))) (0 + (1 + I * ↑y))\n⊢ (fun x ↦ LFunction χ (↑x + (1 + I * ↑y))) =O[𝓝[>] 0] fun x ↦ ↑x",
"ppTerm": "?m.86",
... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\ninst✝ : NeZero N\ny : ℝ\nhy : y ≠ 0 ∨ χ ≠ 1\nh : LFunction χ (1 + I * ↑y) = 0\nthis : HasDerivAt (LFunction χ) (deriv (LFunction χ) (0 + (1 + I * ↑y))) (0 + (1 + I * ↑y))\n⊢ (fun x ↦ LFunction χ (↑x + (1 + I * ↑y))) =O[𝓝[>] 0] fun x ↦ ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 364,
"column": 8
} | {
"line": 364,
"column": 64
} | {
"line": 364,
"column": 65
} | [
{
"pp": "case inr\nN : ℕ\nχ : DirichletCharacter ℂ N\ninst✝ : NeZero N\nt : ℝ\nh✝ : χ ^ 2 ≠ 1 ∨ t ≠ 0\nHz : LFunction χ (1 + I * ↑t) = 0\nhz₁ : t ≠ 0 ∨ χ ≠ 1\nhz₂ : 2 * t ≠ 0 ∨ χ ^ 2 ≠ 1\nx : ℝ\nh : x ≠ 0\n⊢ (↑x ^ 3)⁻¹ * ↑x ^ 3 * ↑x * 1 = ↑x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [
"case inr\nN : ℕ\nχ : DirichletCharacter ℂ N\ninst✝ : NeZero N\nt : ℝ\nh✝ : χ ^ 2 ≠ 1 ∨ t ≠ 0\nHz : LFunction χ (1 + I * ↑t) = 0\nhz₁ : t ≠ 0 ∨ χ ≠ 1\nhz₂ : 2 * t ≠ 0 ∨ χ ^ 2 ≠ 1\nx : ℝ\nh : x ≠ 0\n⊢ 1 * ↑x * 1 = ↑x"
] | inv_mul_cancel₀ <| pow_ne_zero 3 (ofReal_ne_zero.mpr h), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 450,
"column": 8
} | {
"line": 451,
"column": 43
} | {
"line": 451,
"column": 44
} | [
{
"pp": "case refine_1\nq : ℕ\ninst✝ : NeZero q\na : ZMod q\nha : IsUnit a\nH : Summable fun n ↦ (if Nat.Prime n then residueClass a n else 0) / ↑n\nkey : Summable fun n ↦ residueClass a n / ↑n\nC : ℝ := ∑' (n : ℕ), residueClass a n / ↑n\nH₁✝ : ∀ {x : ℝ}, 1 < x → ∑' (n : ℕ), residueClass a n / ↑n ^ x ≤ C\nC' : ... | [
"case refine_1\nq : ℕ\ninst✝ : NeZero q\na : ZMod q\nha : IsUnit a\nH : Summable fun n ↦ (if Nat.Prime n then residueClass a n else 0) / ↑n\nkey : Summable fun n ↦ residueClass a n / ↑n\nC : ℝ := ∑' (n : ℕ), residueClass a n / ↑n\nH₁✝ : ∀ {x : ℝ}, 1 < x → ∑' (n : ℕ), residueClass a n / ↑n ^ x ≤ C\nC' : ℝ\nhC' : ∀ {... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 17
} | {
"line": 91,
"column": 18
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\na : ℤ\nhap : ↑a ≠ 0\n⊢ ↑a ^ (p / 2) * ↑(p / 2)! = ↑((-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val})) * ↑(p / 2)!",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\na : ℤ\nhap : ↑a ≠ 0\n⊢ ↑a ^ (p / 2) = (-1) ^ #({x ∈ Ico 1 (p / 2 + 1) | p / 2 < (↑a * ↑x).val}) ∨ ↑(p / 2)! = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 45
} | {
"line": 499,
"column": 46
} | [
{
"pp": "n q : ℕ\na : ℤ\nhq : q ≠ 0\nh : IsCoprime a ↑q\nthis✝ : NeZero q\nthis : IsUnit ↑a\np : ℕ\nhpn : p > n\nhpp : Prime p\nheq : ↑p = ↑a\n⊢ ↑p ≡ a [ZMOD ↑q]",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"ZMod.commRing... | [
"n q : ℕ\na : ℤ\nhq : q ≠ 0\nh : IsCoprime a ↑q\nthis✝ : NeZero q\nthis : IsUnit ↑a\np : ℕ\nhpn : p > n\nhpp : Prime p\nheq : ↑p = ↑a\n⊢ ↑p = ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 13
} | {
"line": 506,
"column": 14
} | [
{
"pp": "n q a : ℕ\nhq : q ≠ 0\nh : a.Coprime q\n⊢ ∃ p > n, Prime p ∧ p ≡ a [MOD q]",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Prime",
"congrArg",
"Exists",
"id",
"funext",
"GT.gt",
"And",
"Nat.ModEq",
"Nat",
... | [
"n q a : ℕ\nhq : q ≠ 0\nh : a.Coprime q\n⊢ ∃ p, n < p ∧ Prime p ∧ p ≡ a [MOD q]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 139,
"column": 4
} | {
"line": 140,
"column": 50
} | {
"line": 141,
"column": 6
} | [
{
"pp": "p : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact (p % 2 = 1)\na : ℕ\nha2✝ : a % 2 = 1\nhap : ↑a ≠ 0\nha2 : ↑a = ↑1\n⊢ ↑(#({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val})) - ↑(∑ x ∈ Ico 1 (p / 2).succ, x * a / p) = 0",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"AddGrou... | [
"p : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact (p % 2 = 1)\na : ℕ\nha2✝ : a % 2 = 1\nhap : ↑a ≠ 0\nha2 : ↑a = ↑1\n⊢ ↑(#({x ∈ Ico 1 (p / 2 + 1) | p / 2 < (↑a * ↑x).val})) + ∑ x ∈ Ico 1 (p / 2 + 1), ↑(a * x / p) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 204,
"column": 4
} | {
"line": 205,
"column": 28
} | {
"line": 205,
"column": 29
} | [
{
"pp": "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)‖",
"ppTerm": "?m.222",
... | [
"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) / t) ≤ ε * t ^ (-s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 23
} | {
"line": 188,
"column": 24
} | [
{
"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})\nx : ℕ × ℕ\nhx : x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ\nhpq : x.2 * p ≤ x.1 * q\nhqp : x.1... | [
"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})\nx : ℕ × ℕ\nhx : x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ\nhpq : x.2 * p ≤ x.1 * q\nhqp : x.1 * q ≤ x.2 *... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 21
} | {
"line": 453,
"column": 22
} | [
{
"pp": "case inl\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha1 : a % 4 = 1\n⊢ (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
... | [
"case inl\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha1 : a % 4 = 1\n⊢ J(↑b | a) = J(↑a | b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 455,
"column": 4
} | {
"line": 455,
"column": 21
} | {
"line": 455,
"column": 22
} | [
{
"pp": "case inr.inl\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha3 : a % 4 = 3\nhb1 : b % 4 = 1\n⊢ (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"Nat.instAtL... | [
"case inr.inl\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha3 : a % 4 = 3\nhb1 : b % 4 = 1\n⊢ J(↑b | a) = J(↑a | b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 24
} | {
"line": 456,
"column": 25
} | [
{
"pp": "case inr.inr\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha3 : a % 4 = 3\nhb3 : b % 4 = 3\n⊢ (if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)) = J(↑a | b)",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"and_self",
"id",
... | [
"case inr.inr\na b : ℕ\nha2 : a % 2 = 1\nhb2 : b % 2 = 1\nha3 : a % 4 = 3\nhb3 : b % 4 = 3\n⊢ -J(↑b | a) = J(↑a | b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 271,
"column": 4
} | {
"line": 271,
"column": 15
} | {
"line": 271,
"column": 16
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Se... | [
"f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 272,
"column": 2
} | {
"line": 273,
"column": 40
} | {
"line": 274,
"column": 2
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Se... | [
"f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖... | have h₁ : IntegrableOn (fun t ↦ ‖S t - l * t‖ * t ^ (-s - 1)) (Set.Ici 1) :=
lemma₂ hs h₀ fun t ht ↦ (hT t ht).le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 90
} | {
"line": 42,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : ContinuousConstVAdd R R\nh₀ : ∀ (s : Set R), s ∈ 𝓝 0 ↔ ∃ γ, {z | v z < ↑γ} ⊆ s\ns : Set R\nx : R\n⊢ s ∈ 𝓝 x ↔ ∃ γ, (fun x_1 ↦ x + x_1) '' {z | v z < ↑γ} ⊆ s",
"ppTerm": "?m.35",
"assigned": true,
... | [
"R : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : ContinuousConstVAdd R R\nh₀ : ∀ (s : Set R), s ∈ 𝓝 0 ↔ ∃ γ, {z | v z < ↑γ} ⊆ s\ns : Set R\nx : R\n⊢ (fun x_1 ↦ x + x_1) ⁻¹' s ∈ 𝓝 0 ↔ ∃ γ, {a | v (-x + a) < ↑γ} ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 58
} | {
"line": 62,
"column": 6
} | [
{
"pp": "R✝ : Type u_1\ninst✝⁸ : Ring R✝\ninst✝⁷ : ValuativeRel R✝\ninst✝⁶ : TopologicalSpace R✝\ninst✝⁵ : IsValuativeTopology R✝\nR : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : ValuativeRel R\ninst✝² : UniformSpace R\ninst✝¹ : IsUniformAddGroup R\ninst✝ : IsValuativeTopology R\na✝ : Set R\n⊢ failed to pretty print ex... | [
"R✝ : Type u_1\ninst✝⁸ : Ring R✝\ninst✝⁷ : ValuativeRel R✝\ninst✝⁶ : TopologicalSpace R✝\ninst✝⁵ : IsValuativeTopology R✝\nR : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : ValuativeRel R\ninst✝² : UniformSpace R\ninst✝¹ : IsUniformAddGroup R\ninst✝ : IsValuativeTopology R\na✝ : Set R\n⊢ failed to pretty print expression (us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 73,
"column": 4
} | {
"line": 74,
"column": 11
} | {
"line": 74,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : v x = 0\n⊢ Tendsto (⇑v) (𝓝 x) (𝓝 (v x))",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"WithZeroTopology.topologicalSpace",
"Uni... | [
"case pos\nR : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : v x = 0\n⊢ ∀ (ib : ValueGroupWithZero R), ¬ib = 0 → ∃ ia, ∀ (x_1 : R), v.restrict (x_1 - x) < ↑ia → v x_1 < ib"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 76,
"column": 4
} | {
"line": 77,
"column": 11
} | {
"line": 77,
"column": 12
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : ¬v x = 0\n⊢ Tendsto (⇑v) (𝓝 x) (𝓝 (v x))",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"WithZeroTopology.topologicalSpace",
"Un... | [
"case neg\nR : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : ¬v x = 0\n⊢ ∃ ia, ∀ (x_1 : R), v.restrict (x_1 - x) < ↑ia → v x_1 = v x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.ValuativeRel | {
"line": 78,
"column": 19
} | {
"line": 78,
"column": 61
} | {
"line": 78,
"column": 62
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : ¬v x = 0\nx✝ : R\n⊢ v.restrict (x✝ - x) < ↑((Units.mapEquiv ↑(ValueGroupWithZero.orderMonoidIso v)) (Units.mk0 (v x) hx)) → v x✝ = v x",
"ppTerm": "?m.95",
"assigned":... | [
"R : Type u_1\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsValuativeTopology R\nx : R\nhx : ¬v x = 0\nx✝ : R\n⊢ v (x✝ - x) < v x → v x✝ = v x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 36
} | {
"line": 88,
"column": 37
} | [
{
"pp": "K : Type u_1\ninst✝¹ : NontriviallyNormedField K\ninst✝ : IsUltrametricDist K\nx : K\nhx : 0 < ‖x‖\nhx' : ‖x‖ < 1\n⊢ 0 < ‖↑⟨x, ⋯⟩‖ ∧ ‖↑⟨x, ⋯⟩‖ < 1",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",... | [
"K : Type u_1\ninst✝¹ : NontriviallyNormedField K\ninst✝ : IsUltrametricDist K\nx : K\nhx : 0 < ‖x‖\nhx' : ‖x‖ < 1\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 70
} | {
"line": 137,
"column": 4
} | [
{
"pp": "case mp\nK : Type u_1\nΓ₀ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : Valued K Γ₀\ninst✝¹ : v.RankOne\ninst✝ : IsDiscreteValuationRing ↥𝒪[K]\nH : TotallyBounded Set.univ\np : ↥𝒪[K]\nhp : Irreducible p\nthis : ∃ t ⊆ Set.univ, t.Finite ∧ Set.univ ⊆ ⋃ y ∈ t, Metric... | [
"case mp\nK : Type u_1\nΓ₀ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : Valued K Γ₀\ninst✝¹ : v.RankOne\ninst✝ : IsDiscreteValuationRing ↥𝒪[K]\nH : TotallyBounded Set.univ\np : ↥𝒪[K]\nhp : Irreducible p\nthis : ∃ t, t.Finite ∧ ⋃ y ∈ t, Metric.ball y ‖p‖ = Set.univ\n⊢ Finite �... | simp only [Set.subset_univ, Set.univ_subset_iff, true_and] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 296,
"column": 71
} | {
"line": 296,
"column": 82
} | {
"line": 296,
"column": 83
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Se... | [
"f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 83,
"column": 24
} | {
"line": 83,
"column": 35
} | {
"line": 83,
"column": 36
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\nhγ : ¬γ = 0\nthis✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K\nthis : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup\ns : Set K\nhs : s ∈ nhds ... | [
"K : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\nhγ : ¬γ = 0\nthis✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K\nthis : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup\ns : Set K\nhs : s ∈ nhds 0\nhs' : IsC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 86,
"column": 6
} | {
"line": 89,
"column": 41
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\nhγ : ¬γ = 0\nthis✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K\nthis : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup\ns : Set K\nhs :... | [] | refine ⟨r', hr', hr, .trans ?_ hrs⟩
intro x hx
dsimp at hx ⊢
exact hx.trans_lt (hr.trans_le hr1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 86,
"column": 6
} | {
"line": 89,
"column": 41
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\nhγ : ¬γ = 0\nthis✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K\nthis : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup\ns : Set K\nhs :... | [] | refine ⟨r', hr', hr, .trans ?_ hrs⟩
intro x hx
dsimp at hx ⊢
exact hx.trans_lt (hr.trans_le hr1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 228,
"column": 26
} | {
"line": 228,
"column": 44
} | {
"line": 228,
"column": 45
} | [
{
"pp": "K : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc : IsCompact ↑𝒪[K]\nz : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ\na : K\nha : (MonoidWithZeroHom.ofClass v) a = ↑↑z\nhz1 : z ≤ 1\nz0' : 0 < ↑z\nz0 : 0 < ↑↑z\n⊢ 0 < v a",
"p... | [
"K : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc : IsCompact ↑𝒪[K]\nz : (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ\na : K\nha : (MonoidWithZeroHom.ofClass v) a = ↑↑z\nhz1 : z ≤ 1\nz0' : 0 < ↑z\nz0 : 0 < ↑↑z\n⊢ 0 < v a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 21
} | {
"line": 181,
"column": 22
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : ValuativeRel K\ninst✝² : UniformSpace K\ninst✝¹ : IsUniformAddGroup K\ninst✝ : IsNonarchimedeanLocalField K\nf : ℕ → ↥𝒪[K]\nhf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD 𝓂[K] ^ m • ⊤]\nS : ℕ → Set ↥𝒪[K] := ⋯\nn : ℕ\n⊢ f (n + 1) +ᵥ ↑(𝓂[K] ^ n) ⊆ S n",
"ppTerm... | [
"K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : ValuativeRel K\ninst✝² : UniformSpace K\ninst✝¹ : IsUniformAddGroup K\ninst✝ : IsNonarchimedeanLocalField K\nf : ℕ → ↥𝒪[K]\nhf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD 𝓂[K] ^ m • ⊤]\nS : ℕ → Set ↥𝒪[K] := fun n ↦ f n +ᵥ ↑(𝓂[K] ^ n)\nn : ℕ\n⊢ f (n + 1) ≡ f n [SMOD 𝓂[K] ^ n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 32
} | {
"line": 187,
"column": 33
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : ValuativeRel K\ninst✝² : UniformSpace K\ninst✝¹ : IsUniformAddGroup K\ninst✝ : IsNonarchimedeanLocalField K\nf : ℕ → ↥𝒪[K]\nhf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD 𝓂[K] ^ m • ⊤]\nS : ℕ → Set ↥𝒪[K] := fun n ↦ f n +ᵥ ↑(𝓂[K] ^ n)\nhS : ∀ (n : ℕ), S (n + 1) ⊆ ... | [
"K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : ValuativeRel K\ninst✝² : UniformSpace K\ninst✝¹ : IsUniformAddGroup K\ninst✝ : IsNonarchimedeanLocalField K\nf : ℕ → ↥𝒪[K]\nhf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD 𝓂[K] ^ m • ⊤]\nS : ℕ → Set ↥𝒪[K] := fun n ↦ f n +ᵥ ↑(𝓂[K] ^ n)\nhS : ∀ (n : ℕ), S (n + 1) ⊆ S n\nh : ∀ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 32,
"column": 30
} | {
"line": 32,
"column": 59
} | {
"line": 32,
"column": 60
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : IsSimpleRing A\nx : A\nhx1✝ : x ∈ Subring.center A\nhx1 : ∀ (g : A), g * x = x * g\nhx2 : ⟨x, hx1✝⟩ ≠ 0\n⊢ x ≠ 0",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"Zero.toOfNat0",
"OfNat.ofNat",
"Ring.t... | [
"A : Type u_1\ninst✝¹ : Ring A\ninst✝ : IsSimpleRing A\nx : A\nhx1✝ : x ∈ Subring.center A\nhx1 : ∀ (g : A), g * x = x * g\nhx2 : ⟨x, hx1✝⟩ ≠ 0\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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