module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 42
} | [
{
"pp": "s : ℂ\nhs : 0 < s.re\nX : ℝ\nhX : 0 ≤ X\nx : ℝ\nhx : x ∈ Ioo 0 X\nd1 : HasDerivAt (fun y ↦ rexp (-y)) (-rexp (-x)) x\nd2 : HasDerivAt (fun y ↦ ↑y ^ s) (s * ↑x ^ (s - 1)) x\n⊢ HasDerivAt (fun x ↦ ↑(rexp (-x)) * ↑x ^ s) (-(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) x",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 15
} | [
{
"pp": "s : ℂ\nhs : 0 < s.re\nthis✝ : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma\nthis : Tendsto (fun X ↦ -↑X ^ s * ↑(rexp (-X))) atTop (𝓝 0)\n⊢ Tendsto (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) atTop (𝓝 (s * s.GammaIntegral))",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 459,
"column": 33
} | {
"line": 459,
"column": 64
} | [
{
"pp": "E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : FiniteDimensional ℝ F\ninst✝² : MeasurableSpace F\ninst✝¹ : BorelSpace F\nμ : Measure F\ninst✝ : μ.IsAddHaarMeasure\nφ : F → ℝ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 463,
"column": 43
} | {
"line": 463,
"column": 54
} | [
{
"pp": "E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : FiniteDimensional ℝ F\ninst✝² : MeasurableSpace F\ninst✝¹ : BorelSpace F\nμ : Measure F\ninst✝ : μ.IsAddHaarMeasure\nφ : F → ℝ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 314,
"column": 28
} | {
"line": 314,
"column": 66
} | [
{
"pp": "s : ℂ\nh2 : s ≠ 0\nn : ℕ := ⌊1 - s.re⌋₊\n⊢ -s.re < ↑n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 387,
"column": 19
} | {
"line": 387,
"column": 29
} | [
{
"pp": "a : ℂ\nr : ℝ\nha : 0 < a.re\nhr : 0 < r\naux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)\n⊢ ↑r⁻¹ * ∫ (x : ℝ) in Ioi 0, (1 / ↑r) ^ (a - 1) * ↑x ^ (a - 1) * cexp (-↑x) =\n 1 / ↑r * ∫ (t : ℝ) in Ioi 0, (1 / ↑r) ^ (a - 1) * ↑t ^ (a - 1) * cexp (-↑t)",
"usedConstants": [
"instInnerProductSpace... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 44,
"column": 6
} | {
"line": 44,
"column": 17
} | [
{
"pp": "case inl\nr : ℝ\nhr : r ∈ Ioi 0\nhθ : 0 ∈ Ioo (-π) π\n⊢ ((r, 0).1 * cos (r, 0).2, (r, 0).1 * sin (r, 0).2) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}",
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.cos",
"congrArg",
"setOf",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 47,
"column": 6
} | {
"line": 48,
"column": 52
} | [
{
"pp": "case inr.h\nr θ : ℝ\nhθ : θ ∈ Ioo (-π) π\nh'θ : θ ≠ 0\nhr : 0 < r\n⊢ ((r, θ).1 * cos (r, θ).2, (r, θ).1 * sin (r, θ).2) ∈ {q | q.2 ≠ 0}",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"False",
"Real",
"Preorder.toLT",
"Real.pi",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 84
} | [
{
"pp": "⊢ Gamma 0 = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 440,
"column": 2
} | {
"line": 441,
"column": 33
} | [
{
"pp": "n : ℕ\n⊢ Gamma (-↑n) = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 515,
"column": 4
} | {
"line": 515,
"column": 22
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : NormedSpace ℝ V\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 519,
"column": 6
} | {
"line": 519,
"column": 17
} | [
{
"pp": "case succ\nn✝ : ℕ\nn_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0\ns : ℝ\nhs : ∀ (m : ℕ), s ≠ -↑m\nhs' : -↑(n✝ + 1) < s\nthis : Gamma (s + 1) ≠ 0\n⊢ s ≠ 0",
"usedConstants": [
"Real",
"Real.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"OfNat.ofNat"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 84,
"column": 25
} | {
"line": 84,
"column": 41
} | [
{
"pp": "x y : ℝ\nhxy : (x, y) ∈ {q | 0 < q.1} ∪ {q | q.2 ≠ 0}\n⊢ Complex.equivRealProd.symm (x, y) ∈ Complex.slitPlane",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 29
} | [
{
"pp": "s b p : ℝ\nhp : 1 < p\nhb : 0 < b\n⊢ (fun x ↦ x ^ s * rexp (-x)) =o[atTop] fun x ↦ rexp (-(1 / 2) * x)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 90
} | [
{
"pp": "case inl\np s : ℝ\nhs : -1 < s\nhp✝ : 1 ≤ p\nhp : 1 < p\n⊢ IntegrableOn (fun x ↦ x ^ s * rexp (-x ^ p)) (Ioi 0) volume",
"usedConstants": [
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"ContinuousAt",
"PseudoMetricSpace.toUniformSpace",
"continuousAt_neg",
... | have h_exp : ∀ x, ContinuousAt (fun x => exp (-x)) x := fun x => continuousAt_neg.rexp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 217,
"column": 6
} | {
"line": 218,
"column": 53
} | [
{
"pp": "f : ℂ → ℝ≥0∞\n⊢ ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (↑Complex.polarCoord.symm p) = ∫⁻ (p : ℂ), f p",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"MeasurableEquiv.instEquivLike",
"Eq.mpr",
"MeasurableEquiv.measurableEmbedding",
"Normed... | ← (volume_preserving_equiv_real_prod.symm).lintegral_comp_emb
measurableEquivRealProd.symm.measurableEmbedding, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 35
} | [
{
"pp": "b : ℝ\nhb : 0 < b\ns : ℝ\nhs : -1 < s\nthis : MeasurableSet (Ioi 0)\nx : ℝ\nhx : x ∈ Ioi 0\nh'x : 0 ≤ x\n⊢ |(-x) ^ s| ≤ x ^ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 13
} | [
{
"pp": "b : ℝ\nhb : 0 < b\n⊢ Integrable (fun x ↦ rexp (-b * x ^ 2)) volume",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"MeasureTheory.Measure",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Measur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 13
} | [
{
"pp": "b : ℝ\nh : IntegrableOn (fun x ↦ rexp (-b * x ^ 2)) (Ioi 0) volume\nhb : b ≤ 0\nthis : ∫⁻ (x : ℝ) in Ioi 0, 1 ≤ ∫⁻ (x : ℝ) in Ioi 0, ↑‖rexp (-b * x ^ 2)‖₊\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 13
} | [
{
"pp": "b : ℝ\nhb : 0 < b\n⊢ Integrable (fun x ↦ x * rexp (-b * x ^ 2)) volume",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"MeasureTheory.Measure",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Me... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 585,
"column": 4
} | {
"line": 585,
"column": 47
} | [
{
"pp": "E : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 214,
"column": 18
} | {
"line": 214,
"column": 79
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\np : ℝ × ℝ\n| ↑p.1 ^ 2",
"usedConstants": [
"MulOne.toOne",
"Real",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.sin",
"MulOne.toMul",
"instOfNatNat",
"Complex.ofReal",
"Prod.fst",
"MulZeroOneClass.toMulOneCl... | rw [← one_mul ((p.1 : ℂ) ^ 2), ← sin_sq_add_cos_sq (p.2 : ℂ)] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 214,
"column": 18
} | {
"line": 214,
"column": 79
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\np : ℝ × ℝ\n| ↑p.1 ^ 2",
"usedConstants": [
"MulOne.toOne",
"Real",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.sin",
"MulOne.toMul",
"instOfNatNat",
"Complex.ofReal",
"Prod.fst",
"MulZeroOneClass.toMulOneCl... | rw [← one_mul ((p.1 : ℂ) ^ 2), ← sin_sq_add_cos_sq (p.2 : ℂ)] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 214,
"column": 18
} | {
"line": 214,
"column": 79
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\np : ℝ × ℝ\n| ↑p.1 ^ 2",
"usedConstants": [
"MulOne.toOne",
"Real",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.sin",
"MulOne.toMul",
"instOfNatNat",
"Complex.ofReal",
"Prod.fst",
"MulZeroOneClass.toMulOneCl... | rw [← one_mul ((p.1 : ℂ) ^ 2), ← sin_sq_add_cos_sq (p.2 : ℂ)] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 599,
"column": 6
} | {
"line": 599,
"column": 70
} | [
{
"pp": "case h.h\nE : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 620,
"column": 32
} | {
"line": 620,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 620,
"column": 68
} | {
"line": 620,
"column": 79
} | [
{
"pp": "E : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 62
} | [
{
"pp": "case inl\nb : ℝ\nhb : b ≤ 0\n⊢ ¬Integrable (fun x ↦ rexp (-b * x ^ 2)) volume",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"MeasureTheory.MeasureSpace.toMeasurableSpace",
"PartialOrder.toPreorder",
"PseudoMetricSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Pi | {
"line": 147,
"column": 2
} | {
"line": 149,
"column": 45
} | [
{
"pp": "ι : Type u_2\ninst✝³ : Fintype ι\nX : ι → Type u_3\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)\ni : ι\nf : X i → E\nhf : AEStronglyMeasurable f (μ i)\n⊢ ∫ (x : (i... | rw [← (measurePreserving_eval μ i).map_eq, integral_map]
· exact Measurable.aemeasurable (by fun_prop)
· rwa [(measurePreserving_eval μ i).map_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Pi | {
"line": 147,
"column": 2
} | {
"line": 149,
"column": 45
} | [
{
"pp": "ι : Type u_2\ninst✝³ : Fintype ι\nX : ι → Type u_3\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)\ni : ι\nf : X i → E\nhf : AEStronglyMeasurable f (μ i)\n⊢ ∫ (x : (i... | rw [← (measurePreserving_eval μ i).map_eq, integral_map]
· exact Measurable.aemeasurable (by fun_prop)
· rwa [(measurePreserving_eval μ i).map_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 301,
"column": 4
} | {
"line": 301,
"column": 44
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nfull_integral :\n ∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2) ∂volume + ∫ (x : ℝ) in Iic 0, cexp (-b * ↑x ^ 2) ∂volume = (↑π / b) ^ (1 / 2)\nthis✝ : MeasurableSet (Ioi 0)\nc : ℝ\nthis : ∫ (x : ℝ) in 0..c, cexp (-b * ↑(0 - x) ^ 2) = ∫ (x : ℝ) in 0 - c..0 - 0, cexp (-b * ↑x ^ 2)\n⊢ ∫ (x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 348,
"column": 4
} | {
"line": 348,
"column": 56
} | [
{
"pp": "case h.e'_2\n⊢ Gamma (1 / 2) = ↑(Real.Gamma (1 / 2))",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instDivInvMonoid",
"Complex.Gamma",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 370,
"column": 14
} | {
"line": 370,
"column": 45
} | [
{
"pp": "case succ\nk : ℕ\n⊢ Gamma (↑(k + 1) + 1 / 2) = ↑(2 * (k + 1) - 1)‼ * √π / 2 ^ (k + 1)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"Real.instDivInv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 83,
"column": 6
} | {
"line": 87,
"column": 12
} | [
{
"pp": "case a.a.hab.h.hbc\nb : ℂ\nhb : 0 < b.re\nc✝ T✝ : ℝ\nhT✝ : 0 ≤ T✝\nT : ℝ\nhT : 0 ≤ T\nc y : ℝ\nhy : |y| ≤ |c|\n⊢ 2 * b.im * y ≤ 2 * |b.im| * |c|",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Semigroup.toMul",
"Real",
"IsOrderedRin... | (conv_lhs => rw [mul_assoc]); (conv_rhs => rw [mul_assoc])
gcongr _ * ?_
refine (le_abs_self _).trans ?_
rw [abs_mul]
gcongr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 83,
"column": 6
} | {
"line": 87,
"column": 12
} | [
{
"pp": "case a.a.hab.h.hbc\nb : ℂ\nhb : 0 < b.re\nc✝ T✝ : ℝ\nhT✝ : 0 ≤ T✝\nT : ℝ\nhT : 0 ≤ T\nc y : ℝ\nhy : |y| ≤ |c|\n⊢ 2 * b.im * y ≤ 2 * |b.im| * |c|",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Semigroup.toMul",
"Real",
"IsOrderedRin... | (conv_lhs => rw [mul_assoc]); (conv_rhs => rw [mul_assoc])
gcongr _ * ?_
refine (le_abs_self _).trans ?_
rw [abs_mul]
gcongr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.Inversion | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 17
} | [
{
"pp": "V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Inte... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Inversion | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 64
} | [
{
"pp": "V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Inte... | convert! tendsto_integral_cexp_sq_smul this using 4 with c w | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 100,
"column": 6
} | {
"line": 100,
"column": 17
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc T : ℝ\nhT : 0 ≤ T\nvert_norm_bound :\n ∀ {T : ℝ},\n 0 ≤ T →\n ∀ {c y : ℝ},\n |y| ≤ |c| → ‖cexp (-b * (↑T + ↑y * I) ^ 2)‖ ≤ rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))\ny : ℝ\nhy : y ∈ uIoc 0 c\n⊢ |y| ≤ |c|",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Inversion | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 15
} | [
{
"pp": "case h\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 42
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc T : ℝ\nhT : 0 ≤ T\nvert_norm_bound :\n ∀ {T : ℝ},\n 0 ≤ T →\n ∀ {c y : ℝ},\n |y| ≤ |c| → ‖cexp (-b * (↑T + ↑y * I) ^ 2)‖ ≤ rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))\ny : ℝ\nhy : y ∈ uIoc 0 c\nabsy : |(-y)| ≤ |c|\n⊢ ‖cexp (-b * (↑T - ↑y * I) ^ 2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 152,
"column": 4
} | {
"line": 154,
"column": 55
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc : ℝ\nI₁ : ℝ → ℂ := fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)\nHI₁ : I₁ = fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)\nI₂ : ℝ → ℂ := fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)\nI₄ : ℝ → ℂ := fun T ↦ ∫ (y : ℝ) in 0..c, cexp (-b * (↑T + ↑y * I) ^ 2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 791,
"column": 57
} | {
"line": 791,
"column": 85
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\nhf : Integrable f volume\nhf' : Integrable (fun x ↦ x • f x) volume\nw : ℝ\n⊢ Integrable (fun v ↦ ‖v‖ * ‖f v‖) volume",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 795,
"column": 6
} | {
"line": 795,
"column": 75
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\nhf : Integrable f volume\nhf' : Integrable (fun x ↦ x • f x) volume\nw : ℝ\nhf'' : Integrable (fun v ↦ ‖v‖ * ‖f v‖) volume\nL : ℝ →L[ℝ] ℝ →L[ℝ] ℝ := (ContinuousLinearMap.mul ℝ ℝ).flip\nthis : Integrable (fun v ↦ (L v).smul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 828,
"column": 4
} | {
"line": 828,
"column": 46
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\nhf : Integrable f volume\nh'f : Differentiable ℝ f\nhf' : Integrable (deriv f) volume\nx : ℝ\n⊢ Integrable (fun x ↦ fderiv ℝ f x) volume",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 192,
"column": 2
} | {
"line": 193,
"column": 9
} | [
{
"pp": "b : ℂ\nhb : b.re < 0\nc d : ℂ\nhb' : b ≠ 0\nH : ¬Integrable (fun x ↦ cexp (b * ↑x ^ 2 + c * ↑x + d)) volume\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 197,
"column": 27
} | {
"line": 197,
"column": 38
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc d : ℂ\n⊢ (-b).re < 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Left.neg_neg_iff._simp_1",
"Real.instZero",
"instIsLeftCancelAddOfAddLeftReflectLE",
"AddMonoid.toAddZeroClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 847,
"column": 40
} | {
"line": 847,
"column": 51
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\nN : ℕ∞\nn : ℕ\nhf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun x ↦ x ^ n • f x) volume\nhn : ↑n ≤ N\nx : ℝ\nA : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) volume\n⊢ AEStronglyMeasurable f volume",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 66
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc : ℂ\nthis : b ≠ 0\n⊢ (-↑π * b).re < 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"Real.instZero",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 49
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\n⊢ (𝓕 fun x ↦ cexp (-↑π * b * ↑x ^ 2)) = fun t ↦ 1 / b ^ (1 / 2) * cexp (-↑π / b * ↑t ^ 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 289,
"column": 29
} | {
"line": 289,
"column": 40
} | [
{
"pp": "ι : Type u_2\ninst✝ : Fintype ι\nb : ι → ℂ\nhb : ∀ (i : ι), 0 < (b i).re\nc : ι → ℂ\ni : ι\n⊢ (-b i).re < 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Left.neg_neg_iff._simp_1",
"Real.instZero",
"instIsLeftC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 13
} | [
{
"pp": "b : ℂ\nV : Type u_1\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nhb : 0 < b.re\n⊢ ∫ (v : V), cexp (-b * ↑‖v‖ ^ 2) = (↑π / b) ^ (↑(Module.finrank ℝ V) / 2)",
"usedConstants": [
"instInnerProduc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 13
} | [
{
"pp": "b : ℂ\nV : Type u_1\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nhb : 0 < b.re\nw : V\n⊢ 𝓕 (fun v ↦ cexp (-b * ↑‖v‖ ^ 2)) w = (↑π / b) ^ (↑(Module.finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑‖w‖ ^ 2 / b)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 13
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℂ 𝕜 E\nV : Type u_3\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 13
} | [
{
"pp": "E : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℂ E\nV : Type u_3\ninst✝¹⁰ : NormedAddCommGroup V\ninst✝⁹ : InnerProductSpace ℝ V\ninst✝⁸ : FiniteDimensional ℝ V\ninst✝⁷ : MeasurableSpace V\ninst✝⁶ : BorelSpace V\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℂ F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 30
} | [
{
"pp": "E : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℂ E\nV : Type u_3\ninst✝¹⁰ : NormedAddCommGroup V\ninst✝⁹ : InnerProductSpace ℝ V\ninst✝⁸ : FiniteDimensional ℝ V\ninst✝⁷ : MeasurableSpace V\ninst✝⁶ : BorelSpace V\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℂ F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 13
} | [
{
"pp": "E : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℂ E\nV : Type u_3\ninst✝¹⁰ : NormedAddCommGroup V\ninst✝⁹ : InnerProductSpace ℝ V\ninst✝⁸ : FiniteDimensional ℝ V\ninst✝⁷ : MeasurableSpace V\ninst✝⁶ : BorelSpace V\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℂ F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 13
} | [
{
"pp": "V : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace ℝ V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : 𝓢(V, F)\n⊢ ∀ (x : V), ‖(𝓕 f).toBoundedContinuousFunction x‖ ≤ ‖f.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 311,
"column": 8
} | {
"line": 311,
"column": 19
} | [
{
"pp": "V : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace ℝ V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : 𝓢(V, F)\n⊢ ∀ (x : V), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑(𝓕 f)) x‖ ≤ ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 79
} | [
{
"pp": "case a\nV : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : FiniteDimensional ℝ V\ninst✝⁴ : MeasurableSpace V\ninst✝³ : BorelSpace V\nH : Type u_4\ninst✝² : NormedAddCommGroup H\ninst✝¹ : InnerProductSpace ℂ H\ninst✝ : CompleteSpace H\nf : 𝓢(V, H)\n⊢ Complex.ofRealLI ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.Support | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 13
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nF : Type u_6\nV : Type u_10\ninst✝³ : FunLike F α β\ninst✝² : TopologicalSpace α\ninst✝¹ : Zero β\ninst✝ : Zero V\nf : F → V\nx : α\n⊢ x ∈ dsupport f ↔ ∀ (i : Set α), x ∈ i ∧ IsOpen[inst✝²] i → ¬IsVanishingOn f i",
"usedConstants": [
"Eq.mpr",
"Distribution.d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 91
} | [
{
"pp": "d : Type u_1\ninst✝¹ : Fintype d\np : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nhp : p ≠ ∞\n⊢ (span ℂ (range (mFourierLp p))).topologicalClosure = ⊤",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 80
} | [
{
"pp": "case pos\nd : Type u_1\ninst✝ : Fintype d\nm n : d → ℤ\nh : m = n\n⊢ ∫ (x : UnitAddTorus d), (mFourier (n + -m)) x = 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"add_neg_cancel",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 238,
"column": 2
} | {
"line": 239,
"column": 24
} | [
{
"pp": "case neg\nd : Type u_1\ninst✝ : Fintype d\nm n : d → ℤ\nh : ¬m = n\ni : d\nhi : m i ≠ n i\n⊢ ∫ (x : UnitAddCircle), (fourier ((n + -m) i)) x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 55
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\n⊢ HasSum (fun i ↦ mFourierCoeff (↑↑f) i • mFourierLp 2 i) f",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"instHSMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 305,
"column": 2
} | {
"line": 306,
"column": 36
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\n⊢ HasSum (fun i ↦ ‖mFourierCoeff (↑↑f) i‖ ^ 2) (∫ (t : UnitAddTorus d), ‖↑↑f t‖ ^ 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 54
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : C(UnitAddTorus d, ℂ)\nh : Summable (mFourierCoeff ⇑f)\nsum_L2 : HasSum (fun i ↦ mFourierCoeff (⇑f) i • mFourierLp 2 i) ((ContinuousMap.toLp 2 volume ℂ) f)\n⊢ Summable fun a ↦ ‖mFourierCoeff (⇑f) a • mFourier a‖",
"usedConstants": [
"Norm.norm",
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 29
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : C(UnitAddTorus d, ℂ)\nh : Summable (mFourierCoeff ⇑f)\nx : UnitAddTorus d\n⊢ HasSum (fun i ↦ mFourierCoeff (⇑f) i • (mFourier i) x) (f x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 35
} | [
{
"pp": "case h\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : Bore... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 130,
"column": 35
} | {
"line": 130,
"column": 46
} | [
{
"pp": "E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 130,
"column": 80
} | {
"line": 130,
"column": 91
} | [
{
"pp": "E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 216,
"column": 6
} | {
"line": 216,
"column": 62
} | [
{
"pp": "case hf\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : InnerProductSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : BorelSpace E\ninst✝⁸ : NormedAddCommGroup F₁\ninst✝⁷ : NormedSpace ℂ F₁\ninst✝⁶ : NormedAddCommGrou... | exact f.integrable.integrable_convolution B g.integrable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Fourier.Convolution | {
"line": 216,
"column": 6
} | {
"line": 216,
"column": 62
} | [
{
"pp": "case hf\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : InnerProductSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : BorelSpace E\ninst✝⁸ : NormedAddCommGroup F₁\ninst✝⁷ : NormedSpace ℂ F₁\ninst✝⁶ : NormedAddCommGrou... | exact f.integrable.integrable_convolution B g.integrable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.Convolution | {
"line": 216,
"column": 6
} | {
"line": 216,
"column": 62
} | [
{
"pp": "case hf\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : InnerProductSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : BorelSpace E\ninst✝⁸ : NormedAddCommGroup F₁\ninst✝⁷ : NormedSpace ℂ F₁\ninst✝⁶ : NormedAddCommGrou... | exact f.integrable.integrable_convolution B g.integrable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 30
} | [
{
"pp": "case neg\nG : Type u_1\nR : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : RCLike R\ninst✝ : Fintype G\nψ₁ ψ₂ : AddChar G R\nh : ¬ψ₁ = ψ₂\nthis : ψ₂ * ψ₁⁻¹ ≠ 1\n⊢ 𝔼 i, ψ₂ i * (ψ₁ i)⁻¹ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 55
} | [
{
"pp": "G : Type u_1\nR : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : RCLike R\ninst✝ : Fintype G\n⊢ card (AddChar G R) ≤ card G",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Inner | {
"line": 151,
"column": 6
} | {
"line": 151,
"column": 85
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\n𝕜 : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : RCLike 𝕜\nf : κ → ι → 𝕜\nhf : ∀ (k : κ), f k ≠ 0\nhinner : Pairwise fun k₁ k₂ ↦ wInner cWeight (f k₁) (f k₂) = 0\nh✝ : Nonempty ι\n⊢ Pairwise fun k₁ k₂ ↦ wInner 1 (f k₁) (f k₂) = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Inner | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 13
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nw f g : ι → ℝ\nhw : 0 ≤ w\n⊢ |wInner w f g| ≤ wInner w |f| |g|",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 55
} | [
{
"pp": "case h\nE : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℂ F\ninst✝² : CompleteSpace F\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\nf : ↥(... | filter_upwards [g.coeFn_toLp (1 - p⁻¹)⁻¹ μ] with x hg | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 219,
"column": 4
} | {
"line": 224,
"column": 27
} | [
{
"pp": "case h\nE : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedSpace ℂ F\ninst✝⁴ : CompleteSpace F\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst✝¹ : FiniteDimensional ℝ E\nin... | intro g g_smooth g_cpt
have hg₁ : HasCompactSupport (Complex.ofRealCLM ∘ g) := g_cpt.comp_left rfl
have hg₂ : ContDiff ℝ ∞ (Complex.ofRealCLM ∘ g) := by fun_prop
calc
_ = toTemperedDistributionCLM F μ p f (hg₁.toSchwartzMap hg₂) := by simp
_ = _ := by simp [hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 219,
"column": 4
} | {
"line": 224,
"column": 27
} | [
{
"pp": "case h\nE : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedSpace ℂ F\ninst✝⁴ : CompleteSpace F\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst✝¹ : FiniteDimensional ℝ E\nin... | intro g g_smooth g_cpt
have hg₁ : HasCompactSupport (Complex.ofRealCLM ∘ g) := g_cpt.comp_left rfl
have hg₂ : ContDiff ℝ ∞ (Complex.ofRealCLM ∘ g) := by fun_prop
calc
_ = toTemperedDistributionCLM F μ p f (hg₁.toSchwartzMap hg₂) := by simp
_ = _ := by simp [hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FiniteAbelian.Basic | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 13
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing K\ninst✝¹ : Algebra R K\ninst✝ : Module.Finite ℤ R\nA B : Submodule R K\nn : ℕ\nhn : n ≠ 0\nhfg : A.FG\nh : ∀ ⦃x : K⦄, x ∈ map (LinearMap.mulLeft R ↑n) A → x ∈ B\nthis✝ : A.toAddSubgroup.FG\nthis : (AddSubgroup.map (nsmulAddMonoidHom n)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 76
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nx y : ℤ\nh : zmod n ↑x = zmod n ↑y\nhn : ↑n ≠ 0\n⊢ ↑x = ↑y",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"ZMod.commRing",
"id",
"Int",
"AddGroupWithOne.toIntCast",
"Nat.cast",
"ZMod",
"CharP.intCast_eq_intCast",
"ZMod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 86,
"column": 50
} | {
"line": 86,
"column": 74
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : DecidableEq ι\nn : ι → ℕ\ninst✝ : ∀ (i : ι), NeZero (n i)\nf g : (i : ι) → ZMod (n i)\nh : (fun i ↦ zmod (n i) (f i)) = fun i ↦ zmod (n i) (g i)\n⊢ f = g",
"usedConstants": [
"Eq.mpr",
"id",
"ZMod",
"_private.Mathlib.Analysis.Fourier.FiniteAbelian.Pont... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddCommGroup α\na : α\ninst✝ : Finite α\n⊢ (∀ (ψ : AddChar α ℂ), ψ a = 1) ↔ a = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 156,
"column": 38
} | {
"line": 156,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddCommGroup α\ninst✝ : Finite α\na : α\nha : a ∈ doubleDualEmb.ker\nψ : AddChar α ℂ\n⊢ ψ a = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\na : α\n⊢ ∑ ψ, ψ a = if a = 0 then ↑(Fintype.card α) else 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : Finite α\ninst✝ : DecidableEq α\na : α\n⊢ 𝔼 ψ, ψ a = if a = 0 then 1 else 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 13
} | [
{
"pp": "case h\nE : Type u_3\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : NormedSpace ℂ F\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\ninst✝ : CompleteSpace F\nf : 𝓢(E, F)\ng : 𝓢(E, ℂ)\n⊢ (𝓕 ((toT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 30
} | [
{
"pp": "case h\nf : C(ℝ, ℂ)\nhf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖\nm : ℤ\ne : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }\nneK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e *... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 70,
"column": 2
} | {
"line": 73,
"column": 12
} | [
{
"pp": "n : ℕ\n⊢ μ n ≠ 0 ↔ Squarefree n",
"usedConstants": [
"Int.instAddCommGroup",
"NegZeroClass.toNeg",
"False",
"Mathlib.Tactic.Contrapose.contrapose₂",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"ArithmeticFunction.instFunLikeNat",
"eq_false",... | constructor <;> intro h
· contrapose h
simp [h]
· simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 70,
"column": 2
} | {
"line": 73,
"column": 12
} | [
{
"pp": "n : ℕ\n⊢ μ n ≠ 0 ↔ Squarefree n",
"usedConstants": [
"Int.instAddCommGroup",
"NegZeroClass.toNeg",
"False",
"Mathlib.Tactic.Contrapose.contrapose₂",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"ArithmeticFunction.instFunLikeNat",
"eq_false",... | constructor <;> intro h
· contrapose h
simp [h]
· simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 12
} | [
{
"pp": "n m : ℕ\nhn : n ≠ 0\nhm : m ≠ 0\nhnm : n.Coprime m\n⊢ μ (n * m) = μ n * μ m",
"usedConstants": [
"Nat.instMulZeroClass",
"ite_zero_mul_ite_zero",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"MulZeroClass.toMul",
"Monoid.toMulOneClass",
"congrArg",
... | simp only [moebius, coe_mk, squarefree_mul hnm, ite_zero_mul_ite_zero, cardFactors_mul hn hm,
pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 113,
"column": 4
} | {
"line": 115,
"column": 12
} | [
{
"pp": "case h.e'_2\nf : C(ℝ, ℂ)\nh_norm : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖\nh_sum : Summable fun n ↦ 𝓕 ⇑f ↑n\nx : ℝ\nF : C(UnitAddCircle, ℂ) := { toFun := ⋯.lift, continuous_toFun := ⋯ }\nthis : Summable (fourierCoeff ⇑F)\n⊢ ∑' (n : ℤ), f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 231,
"column": 6
} | {
"line": 232,
"column": 18
} | [
{
"pp": "case h.succ\nR : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\nf' : ArithmeticFunction R := { toFun := fun x ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }\ng' : ArithmeticFunction R := { toFun := fun x ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }\nn : ℕ\n⊢ (μ • g') (n + 1) = f' (n + 1) ↔ n + 1 > 0 → ∑ ... | simp only [forall_prop_of_true, succ_pos', smul_apply, f', g', coe_mk, succ_ne_zero,
ite_false] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nP : ℕ → Prop\nn : ℕ\nh : n > 0 → n ∈ s → P n\nhn : n ∈ s\nhs₀ : n ≤ 0\n⊢ 0 ∈ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 311,
"column": 2
} | {
"line": 311,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nhs₀ : 0 ∉ s\nthis : ∀ (P : ℕ → Prop), (∀ n ∈ s, P n) ↔ ∀ n > 0, n ∈ s → P n\n⊢ (∀ n ∈ s, ∑ i ∈ n.divisors, f i = g n) ↔ ∀ n ∈ s, ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 66
} | [
{
"pp": "E : Type u_1\nV : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nf : V → E\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nhf1 :\n Continuous[PseudoMetricSpace.toUniformSpace.toTopolog... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.Complex | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 42
} | [
{
"pp": "n : ℕ\nh0 : n ≠ 0\n⊢ IsPrimitiveRoot (cexp (2 * ↑π * I / ↑n)) n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.Complex | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 22
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nx : ℂˣ\nhn0 : ↑n ≠ 0\nh : ↑x ^ n = 1\n⊢ ∃ i < n, cexp (2 * ↑π * I / ↑n) ^ i = ↑x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 316,
"column": 29
} | {
"line": 316,
"column": 54
} | [
{
"pp": "M₀ : Type u_7\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : Nontrivial M₀\nl : ℕ\nhl : 0 ^ l = 1\n⊢ 0 ∣ l",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Nat.instSemigroupWithZero",
"SemigroupWithZero.toMulZeroClass",
"id",
"instOfNatNat",
"Nat.instDvd",
"Nat",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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