module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 584,
"column": 4
} | {
"line": 584,
"column": 47
} | {
"line": 584,
"column": 48
} | [
{
"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✝... | [
"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✝¹ : FiniteDi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1198,
"column": 6
} | {
"line": 1198,
"column": 72
} | {
"line": 1198,
"column": 73
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulC... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1249,
"column": 30
} | {
"line": 1250,
"column": 65
} | {
"line": 1250,
"column": 66
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : ProperSpace E\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedS... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : ProperSpace E\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 598,
"column": 6
} | {
"line": 598,
"column": 70
} | {
"line": 598,
"column": 71
} | [
{
"pp": "case 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 : BorelSpace ... | [
"case 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 : BorelSpace V\ninst✝¹ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 459,
"column": 33
} | {
"line": 459,
"column": 64
} | {
"line": 459,
"column": 65
} | [
{
"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... | [
"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\nhφ : ∀ (x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1313,
"column": 2
} | {
"line": 1313,
"column": 13
} | {
"line": 1313,
"column": 14
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\nμ : Measure E\nC : ℝ\nleft✝ : 0 < C\nhC : ∀... | [
"E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\nμ : Measure E\nC : ℝ\nleft✝ : 0 < C\nhC : ∀ (x : E), ‖x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 463,
"column": 43
} | {
"line": 463,
"column": 54
} | {
"line": 463,
"column": 55
} | [
{
"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... | [
"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\nhφ : ∀ (x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1338,
"column": 17
} | {
"line": 1338,
"column": 89
} | {
"line": 1339,
"column": 4
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\np : ℝ≥0∞\nμ : Measure E\nhp₁ : p ≠ 0\nhp₂ :... | [
"E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\np : ℝ≥0∞\nμ : Measure E\nhp₁ : p ≠ 0\nhp₂ : p ≠ ∞\nhμ :... | MeasureTheory.MemLp.eLpNorm_eq_integral_rpow_norm hp₁ hp₂ (f.memLp p μ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1343,
"column": 2
} | {
"line": 1343,
"column": 13
} | {
"line": 1343,
"column": 14
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\nμ : Measure E\nhμ : μ.HasTemperateGrowth\n⊢... | [
"E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\nμ : Measure E\nhμ : μ.HasTemperateGrowth\n⊢ ‖f.toLp 1 μ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 619,
"column": 32
} | {
"line": 619,
"column": 43
} | {
"line": 619,
"column": 44
} | [
{
"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✝... | [
"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✝¹ : FiniteDi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1353,
"column": 15
} | {
"line": 1353,
"column": 33
} | {
"line": 1353,
"column": 34
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : MeasurableSpace E\ninst✝² : OpensMeasurableSpace E\ninst✝¹ : SecondCountableTopologyEither E F\np : ℝ≥0∞\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst... | [
"E : Type u_5\nF : Type u_6\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : MeasurableSpace E\ninst✝² : OpensMeasurableSpace E\ninst✝¹ : SecondCountableTopologyEither E F\np : ℝ≥0∞\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst✝ : μ.IsOpen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 619,
"column": 68
} | {
"line": 619,
"column": 79
} | {
"line": 619,
"column": 80
} | [
{
"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✝... | [
"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✝¹ : FiniteDi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1365,
"column": 2
} | {
"line": 1365,
"column": 13
} | {
"line": 1365,
"column": 14
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : OpensMeasurableSpace E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : SMulCommClass ℝ 𝕜 F\nin... | [
"𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : OpensMeasurableSpace E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : SMulCommClass ℝ 𝕜 F\ninst✝ : Second... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Integral | {
"line": 64,
"column": 4
} | {
"line": 65,
"column": 27
} | {
"line": 65,
"column": 28
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n ↦ ‖ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖\nn : ℤ\nx : ↑(Icc (↑n) (↑n + 1))\nthis :\n ‖(ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))) ⟨↑x - ↑n, ⋯⟩‖ ≤\n ... | [
"case refine_1\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nhf : Summable fun n ↦ ‖ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))‖\nn : ℤ\nx : ↑(Icc (↑n) (↑n + 1))\nthis :\n ‖(ContinuousMap.restrict (Icc 0 1) (f.comp (ContinuousMap.addRight ↑n))) ⟨↑x - ↑n, ⋯⟩‖ ≤\n ‖Continuous... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1387,
"column": 2
} | {
"line": 1387,
"column": 82
} | {
"line": 1388,
"column": 2
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : OpensMeasurableSpace E\ninst✝³ : SecondCountableTopologyEither E F\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : BorelSpace E\np... | [
"E : Type u_5\nF : Type u_6\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : OpensMeasurableSpace E\ninst✝³ : SecondCountableTopologyEither E F\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : BorelSpace E\np : ℝ≥0∞\nhp ... | refine (mem_closure_iff_nhds_basis Metric.nhds_basis_closedBall).2 fun ε hε ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Lebesgue.Integral | {
"line": 90,
"column": 75
} | {
"line": 92,
"column": 21
} | {
"line": 94,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nc : ℝ\nf : ℝ → E\n⊢ ∫ (x : ℝ) in Ioi c, f (-x) = ∫ (x : ℝ) in Iic (-c), f x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"MeasureTheory.Measure",
"No... | [] | by
rw [← neg_neg c, ← integral_comp_neg_Iic]
simp only [neg_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 42
} | {
"line": 58,
"column": 43
} | [
{
"pp": "c : ℝ\n⊢ ∫ (x : ℝ) in Ioi c, rexp (-x) = rexp (-c)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Set.Ioi",
"Real.instRCLike",
"congrArg",
"MeasureTheory.MeasureSpace.toMeasurableSpace... | [
"c : ℝ\n⊢ ∫ (x : ℝ) in Iic (-c), rexp x = rexp (-c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 39
} | {
"line": 61,
"column": 40
} | [
{
"pp": "⊢ ∫ (x : ℝ) in Ioi 0, rexp (-x) = 1",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ ∫ (x : ℝ) in Ioi 0, rexp (-x) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 13
} | {
"line": 74,
"column": 14
} | [
{
"pp": "a : ℂ\nha : 0 < a.re\nc : ℝ\n⊢ IntegrableOn (fun x ↦ Complex.exp (a * ↑x)) (Iic c) volume",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : ℂ\nha : 0 < a.re\nc : ℝ\n⊢ IntegrableOn (fun x ↦ Complex.exp (a * ↑x)) (Iic c) volume"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 93
} | {
"line": 80,
"column": 2
} | [
{
"pp": "a : ℝ\nha : a < 0\nc : ℝ\n⊢ IntegrableOn (fun x ↦ rexp (a * x)) (Ioi c) volume",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Set.Ioi",
"HMul.hMul",
"Complex.instNormedAddCommGroup",
... | [
"a : ℝ\nha : a < 0\nc : ℝ\nthis : Integrable (fun a_1 ↦ ‖Complex.exp (↑a * ↑a_1)‖) (volume.restrict (Ioi c))\n⊢ IntegrableOn (fun x ↦ rexp (a * x)) (Ioi c) volume"
] | have := Integrable.norm <| integrableOn_exp_mul_complex_Ioi (a := a) (by simpa using! ha) c | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 15
} | {
"line": 93,
"column": 16
} | [
{
"pp": "a : ℂ\nha : a.re < 0\nc : ℝ\nthis : Tendsto (fun x ↦ Complex.exp (a * ↑x)) atTop (𝓝 0)\n⊢ Tendsto (fun i ↦ (Complex.exp (a * ↑i) - Complex.exp (a * ↑c)) / a) atTop (𝓝 (-Complex.exp (a * ↑c) / a))",
"ppTerm": "?m.69",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"a : ℂ\nha : a.re < 0\nc : ℝ\nthis : Tendsto (fun x ↦ Complex.exp (a * ↑x)) atTop (𝓝 0)\n⊢ Tendsto (fun i ↦ (Complex.exp (a * ↑i) - Complex.exp (a * ↑c)) / a) atTop (𝓝 (-Complex.exp (a * ↑c) / a))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 49
} | {
"line": 94,
"column": 50
} | [
{
"pp": "a : ℂ\nha : a.re < 0\nc : ℝ\n⊢ Tendsto (fun x ↦ Complex.exp (a * ↑x)) atTop (𝓝 0)",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Complex.mul_re",
"HMul.hMul",
"congrArg",
"sub_zer... | [
"a : ℂ\nha : a.re < 0\nc : ℝ\n⊢ Tendsto (fun x ↦ a.re * x) atTop atBot"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 98,
"column": 2
} | {
"line": 100,
"column": 9
} | {
"line": 100,
"column": 10
} | [
{
"pp": "a : ℂ\nha : 0 < a.re\nc : ℝ\n⊢ ∫ (x : ℝ) in Iic c, Complex.exp (a * ↑x) = Complex.exp (a * ↑c) / a",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : ℂ\nha : 0 < a.re\nc : ℝ\n⊢ ∫ (x : ℝ) in Iic c, Complex.exp (a * ↑x) = Complex.exp (a * ↑c) / a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 105,
"column": 52
} | {
"line": 105,
"column": 63
} | {
"line": 105,
"column": 64
} | [
{
"pp": "a : ℝ\nha : a < 0\nc : ℝ\n⊢ (↑a).re < 0",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Real.instLT",
"id",
"Complex.ofReal",
"Complex.re",
"LT.lt",
"Zero.toOfNat0",
"OfNat.ofNat"
],
"usedFVars... | [
"a : ℝ\nha : a < 0\nc : ℝ\n⊢ a < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 107,
"column": 45
} | {
"line": 107,
"column": 56
} | {
"line": 107,
"column": 57
} | [
{
"pp": "a : ℝ\nha : a < 0\nc : ℝ\n⊢ (↑a).re < 0",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Real.instLT",
"id",
"Complex.ofReal",
"Complex.re",
"LT.lt",
"Zero.toOfNat0",
"OfNat.ofNat"
],
"usedFVars... | [
"a : ℝ\nha : a < 0\nc : ℝ\n⊢ a < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 111,
"column": 2
} | {
"line": 112,
"column": 9
} | {
"line": 112,
"column": 10
} | [
{
"pp": "a : ℝ\nha : 0 < a\nc : ℝ\n⊢ ∫ (x : ℝ) in Iic c, rexp (a * x) = rexp (a * c) / a",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : ℝ\nha : 0 < a\nc : ℝ\n⊢ ∫ (x : ℝ) in Iic c, rexp (a * x) = rexp (a * c) / a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 13
} | {
"line": 132,
"column": 14
} | [
{
"pp": "a c : ℝ\nha : a < -1\nhc : 0 < c\n⊢ IntegrableOn (fun t ↦ t ^ a) (Ioi c) volume",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a c : ℝ\nha : a < -1\nhc : 0 < c\n⊢ IntegrableOn (fun t ↦ t ^ a) (Ioi c) volume"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 30
} | {
"line": 180,
"column": 31
} | [
{
"pp": "a : ℝ\nha : a < -1\nc : ℝ\nhc : 0 < c\nhd : ∀ x ∈ Ici c, HasDerivAt (fun t ↦ t ^ (a + 1) / (a + 1)) (x ^ a) x\n⊢ Tendsto (fun a_1 ↦ a_1 ^ (a + 1)) atTop (𝓝 0)",
"ppTerm": "?m.483",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : ℝ\nha : a < -1\nc : ℝ\nhc : 0 < c\nhd : ∀ x ∈ Ici c, HasDerivAt (fun t ↦ t ^ (a + 1) / (a + 1)) (x ^ a) x\n⊢ Tendsto (fun a_1 ↦ a_1 ^ (a + 1)) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 174,
"column": 2
} | {
"line": 182,
"column": 41
} | {
"line": 184,
"column": 0
} | [
{
"pp": "a : ℝ\nha : a < -1\nc : ℝ\nhc : 0 < c\n⊢ ∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instIsOrderedRing",... | [] | have hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
intro x hx
convert! (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1
simp [show a + 1 ≠ 0 from ne_of_lt (by linarith), mul_comm]
have ht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 174,
"column": 2
} | {
"line": 182,
"column": 41
} | {
"line": 184,
"column": 0
} | [
{
"pp": "a : ℝ\nha : a < -1\nc : ℝ\nhc : 0 < c\n⊢ ∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instIsOrderedRing",... | [] | have hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by
intro x hx
convert! (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1
simp [show a + 1 ≠ 0 from ne_of_lt (by linarith), mul_comm]
have ht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Asymptotics | {
"line": 174,
"column": 21
} | {
"line": 174,
"column": 49
} | {
"line": 174,
"column": 50
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\nf : α → E\ng : α → F\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : SecondCountableTopology α\ninst✝⁸ : MeasurableSpace α\nμ : Measure α\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : AddCommGroup α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsOrdered... | [
"α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\nf : α → E\ng : α → F\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : SecondCountableTopology α\ninst✝⁸ : MeasurableSpace α\nμ : Measure α\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : AddCommGroup α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsOrderedAddMonoid α\... | ← Measure.map_neg_eq_self μ, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.MeasureTheory.Integral.ExpDecay | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 21
} | {
"line": 37,
"column": 22
} | [
{
"pp": "a b : ℝ\nh : 0 < b\nthis : Tendsto (fun x ↦ -rexp (-b * x) / b) atTop (𝓝 (-0 / b))\nx : ℝ\nx✝ : x ∈ Ici a\n⊢ HasDerivAt (fun x ↦ -rexp (-b * x) / b) (rexp (-b * x)) x",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Eq.mpr",
... | [
"a b : ℝ\nh : 0 < b\nthis : Tendsto (fun x ↦ -rexp (-b * x) / b) atTop (𝓝 (-0 / b))\nx : ℝ\nx✝ : x ∈ Ici a\n⊢ HasDerivAt (fun x ↦ -rexp (-(b * x)) / b) (rexp (-(b * x))) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransform | {
"line": 594,
"column": 2
} | {
"line": 594,
"column": 13
} | {
"line": 594,
"column": 14
} | [
{
"pp": "V : Type u_1\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\ninst✝ : FiniteDimensional ℝ V\nf : V → E\nhf : MemLp f 1 volume\nx : V\n⊢ (fourierTransformInv (MemLp.to... | [
"V : Type u_1\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\ninst✝ : FiniteDimensional ℝ V\nf : V → E\nhf : MemLp f 1 volume\nx : V\n⊢ 𝓕⁻ (↑↑(MemLp.toLp f hf)) x = 𝓕⁻ f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 52
} | {
"line": 115,
"column": 2
} | [
{
"pp": "s : ℂ\n⊢ ((starRingEnd ℂ) s).GammaIntegral = (starRingEnd ℂ) s.GammaIntegral",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real"... | [
"s : ℂ\n⊢ ∫ (x : ℝ) in Ioi 0, ↑(rexp (-x)) * ↑x ^ ((starRingEnd ℂ) s - 1) =\n ∫ (x : ℝ) in Ioi 0, (starRingEnd ℂ) (↑(rexp (-x)) * ↑x ^ (s - 1))"
] | rw [GammaIntegral, GammaIntegral, ← integral_conj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 133,
"column": 2
} | {
"line": 134,
"column": 18
} | {
"line": 134,
"column": 19
} | [
{
"pp": "⊢ GammaIntegral 1 = 1",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instPow",
"Real",
"Set.Ioi",
"HMul.hMul",
"sub_self",
"Real.instZero",
"Real.instRCLike",
"congrArg",... | [
"⊢ ∫ (x : ℝ) in Ioi 0, rexp (-x) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 29
} | {
"line": 56,
"column": 30
} | [
{
"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)",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Rea... | [
"s b p : ℝ\nhp : 1 < p\nhb : 0 < b\n⊢ (fun x ↦ x ^ s * rexp (-x)) =o[atTop] fun x ↦ rexp (x * -(1 / 2))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 813,
"column": 40
} | {
"line": 813,
"column": 51
} | {
"line": 813,
"column": 52
} | [
{
"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",
"ppTerm": "?m.... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 35
} | {
"line": 114,
"column": 36
} | [
{
"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",
"ppTerm": "?m.289",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 13
} | {
"line": 118,
"column": 14
} | [
{
"pp": "b : ℝ\nhb : 0 < b\n⊢ Integrable (fun x ↦ rexp (-b * x ^ 2)) volume",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"MeasureTheory.Measure",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [
"b : ℝ\nhb : 0 < b\n⊢ Integrable (fun x ↦ rexp (-(b * x ^ 2))) volume"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | {
"line": 129,
"column": 14
} | [
{
"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",
"ppTerm": "?m.126",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 13
} | {
"line": 137,
"column": 14
} | [
{
"pp": "b : ℝ\nhb : 0 < b\n⊢ Integrable (fun x ↦ x * rexp (-b * x ^ 2)) volume",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"MeasureTheory.Measure",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [
"b : ℝ\nhb : 0 < b\n⊢ Integrable (fun x ↦ x * rexp (-(b * x ^ 2))) volume"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 217,
"column": 6
} | {
"line": 218,
"column": 53
} | {
"line": 218,
"column": 54
} | [
{
"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",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"MeasurableEquiv.instEquivLike",
"Eq.mpr",
"Measur... | [
"f : ℂ → ℝ≥0∞\n⊢ ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (↑Complex.polarCoord.symm p) =\n ∫⁻ (a : ℝ × ℝ), f (measurableEquivRealProd.symm a) ∂MeasureTheory.volume"
] | ← (volume_preserving_equiv_real_prod.symm).lintegral_comp_emb
measurableEquivRealProd.symm.measurableEmbedding, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Pi | {
"line": 147,
"column": 2
} | {
"line": 149,
"column": 45
} | {
"line": 151,
"column": 0
} | [
{
"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
} | {
"line": 151,
"column": 0
} | [
{
"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": 202,
"column": 18
} | {
"line": 202,
"column": 79
} | {
"line": 203,
"column": 6
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\np : ℝ × ℝ\n| ↑p.1 ^ 2",
"ppTerm": "?m.581",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Real",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.sin",
"MulOne.toMul",
"instOfNatNat",
"Complex.ofReal",
... | [
"b : ℂ\nhb : 0 < b.re\np : ℝ × ℝ\n| (Complex.sin ↑p.2 ^ 2 + Complex.cos ↑p.2 ^ 2) * ↑p.1 ^ 2"
] | 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": 202,
"column": 18
} | {
"line": 202,
"column": 79
} | {
"line": 203,
"column": 6
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\np : ℝ × ℝ\n| ↑p.1 ^ 2",
"ppTerm": "?m.581",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Real",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.sin",
"MulOne.toMul",
"instOfNatNat",
"Complex.ofReal",
... | [
"b : ℂ\nhb : 0 < b.re\np : ℝ × ℝ\n| (Complex.sin ↑p.2 ^ 2 + Complex.cos ↑p.2 ^ 2) * ↑p.1 ^ 2"
] | 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": 202,
"column": 18
} | {
"line": 202,
"column": 79
} | {
"line": 203,
"column": 6
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\np : ℝ × ℝ\n| ↑p.1 ^ 2",
"ppTerm": "?m.581",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Real",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.sin",
"MulOne.toMul",
"instOfNatNat",
"Complex.ofReal",
... | [
"b : ℂ\nhb : 0 < b.re\np : ℝ × ℝ\n| (Complex.sin ↑p.2 ^ 2 + Complex.cos ↑p.2 ^ 2) * ↑p.1 ^ 2"
] | 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.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 216,
"column": 6
} | {
"line": 216,
"column": 62
} | {
"line": 216,
"column": 63
} | [
{
"pp": "case inl\nb : ℝ\nhb : b ≤ 0\n⊢ ¬Integrable (fun x ↦ rexp (-b * x ^ 2)) volume",
"ppTerm": "?inl✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"MeasureTheory.MeasureSpace.toMeasurableSpace",
"Part... | [
"case inl\nb : ℝ\nhb : b ≤ 0\n⊢ b ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 83,
"column": 6
} | {
"line": 87,
"column": 12
} | {
"line": 88,
"column": 4
} | [
{
"pp": "case 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|",
"ppTerm": "?hbc✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Semigroup.toMul"... | [] | (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
} | {
"line": 88,
"column": 4
} | [
{
"pp": "case 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|",
"ppTerm": "?hbc✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Semigroup.toMul"... | [] | (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.SpecialFunctions.Gamma.Basic | {
"line": 314,
"column": 28
} | {
"line": 314,
"column": 66
} | {
"line": 314,
"column": 67
} | [
{
"pp": "s : ℂ\nh2 : s ≠ 0\nn : ℕ := ⌊1 - s.re⌋₊\n⊢ -s.re < ↑n",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : ℂ\nh2 : s ≠ 0\nn : ℕ := ⌊1 - s.re⌋₊\n⊢ -s.re < ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Inversion | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 64
} | {
"line": 87,
"column": 4
} | [
{
"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... | [
"case e'_3\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'f : Integ... | 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
} | {
"line": 100,
"column": 18
} | [
{
"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|",
"ppTerm": "?m.477",
"assigned"... | [
"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|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 387,
"column": 19
} | {
"line": 387,
"column": 29
} | {
"line": 387,
"column": 30
} | [
{
"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)",
"ppTerm": "?m.461",
"assigned": true,
"... | [
"a : ℂ\nr : ℝ\nha : 0 < a.re\nhr : 0 < r\naux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)\n⊢ ↑(1 / r) * ∫ (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)"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 282,
"column": 6
} | {
"line": 282,
"column": 21
} | {
"line": 282,
"column": 22
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nf : ℝ → ℂ := fun x ↦ cexp (-b * ↑x ^ 2)\nfull_integral : ∫ (x : ℝ), cexp (-b * ↑x ^ 2) = (↑π / b) ^ (1 / 2)\n⊢ ∫ (x : ℝ) in Iic 0, f x = ∫ (x : ℝ) in Ioi 0, f (-x)",
"ppTerm": "?m.133",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
... | [
"b : ℂ\nhb : 0 < b.re\nf : ℝ → ℂ := fun x ↦ cexp (-b * ↑x ^ 2)\nfull_integral : ∫ (x : ℝ), cexp (-b * ↑x ^ 2) = (↑π / b) ^ (1 / 2)\n⊢ ∫ (x : ℝ) in Iic 0, cexp (-(b * ↑x ^ 2)) = ∫ (x : ℝ) in Ioi 0, cexp (-(b * ↑x ^ 2))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 84
} | {
"line": 435,
"column": 4
} | [
{
"pp": "⊢ Gamma 0 = 0",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ Gamma 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 440,
"column": 2
} | {
"line": 441,
"column": 33
} | {
"line": 441,
"column": 34
} | [
{
"pp": "n : ℕ\n⊢ Gamma (-↑n) = 0",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\n⊢ Gamma (-↑n) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 287,
"column": 39
} | {
"line": 287,
"column": 50
} | {
"line": 287,
"column": 51
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nf : ℝ → ℂ := fun x ↦ cexp (-b * ↑x ^ 2)\nfull_integral : (∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2)) * 2 = (↑π / b) ^ (1 / 2)\nh_eq : ∫ (x : ℝ) in Iic 0, f x = ∫ (x : ℝ) in Ioi 0, f x\n⊢ (∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2)) * 2 = (↑π / b) ^ (1 / 2)",
"ppTerm": "?m.249",
"... | [
"b : ℂ\nhb : 0 < b.re\nf : ℝ → ℂ := fun x ↦ cexp (-b * ↑x ^ 2)\nfull_integral : (∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2)) * 2 = (↑π / b) ^ (1 / 2)\nh_eq : ∫ (x : ℝ) in Iic 0, f x = ∫ (x : ℝ) in Ioi 0, f x\n⊢ (∫ (x : ℝ) in Ioi 0, cexp (-(b * ↑x ^ 2))) * 2 = (↑π / b) ^ 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 152,
"column": 4
} | {
"line": 154,
"column": 55
} | {
"line": 154,
"column": 56
} | [
{
"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... | [
"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)\nI₅ : ℝ → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 324,
"column": 4
} | {
"line": 324,
"column": 56
} | {
"line": 324,
"column": 57
} | [
{
"pp": "case e'_2\n⊢ Gamma (1 / 2) = ↑(Real.Gamma (1 / 2))",
"ppTerm": "?e'_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instD... | [
"case e'_2\n⊢ Gamma 2⁻¹ = ↑(Real.Gamma 2⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 346,
"column": 14
} | {
"line": 346,
"column": 45
} | {
"line": 346,
"column": 46
} | [
{
"pp": "case succ\nk : ℕ\n⊢ Gamma (↑(k + 1) + 1 / 2) = ↑(2 * (k + 1) - 1)‼ * √π / 2 ^ (k + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"AddMonoid.toAddSemigro... | [
"case succ\nk : ℕ\n⊢ Gamma (↑k + 1 + 1 / 2) = ↑(2 * k + 1)‼ * √π / 2 ^ (k + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 520,
"column": 6
} | {
"line": 520,
"column": 17
} | {
"line": 520,
"column": 18
} | [
{
"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",
"ppTerm": "?succ✝",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"id",
"Ne",... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 192,
"column": 2
} | {
"line": 193,
"column": 9
} | {
"line": 193,
"column": 10
} | [
{
"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",
"ppTerm": "?m.73",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 197,
"column": 27
} | {
"line": 197,
"column": 38
} | {
"line": 197,
"column": 39
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc d : ℂ\n⊢ (-b).re < 0",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Left.neg_neg_iff._simp_1",
"Real.instZero",
"instIsLeftCancelAddOfAddLeftR... | [
"b : ℂ\nhb : 0 < b.re\nc d : ℂ\n⊢ 0 < b.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 66
} | {
"line": 212,
"column": 67
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc : ℂ\nthis : b ≠ 0\n⊢ (-↑π * b).re < 0",
"ppTerm": "?m.152",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
... | [
"b : ℂ\nhb : 0 < b.re\nc : ℂ\nthis : b ≠ 0\n⊢ 0 < π * b.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 49
} | {
"line": 230,
"column": 50
} | [
{
"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)",
"ppTerm": "?m.94",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 283,
"column": 29
} | {
"line": 283,
"column": 40
} | {
"line": 283,
"column": 41
} | [
{
"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",
"ppTerm": "?m.198",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Left.neg_neg_iff._simp... | [
"ι : Type u_2\ninst✝ : Fintype ι\nb : ι → ℂ\nhb : ∀ (i : ι), 0 < (b i).re\nc : ι → ℂ\ni : ι\n⊢ 0 < (b i).re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 13
} | {
"line": 326,
"column": 14
} | [
{
"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)",
"ppTerm": "?m.66",
"assigned": true,
... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 13
} | {
"line": 355,
"column": 14
} | [
{
"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)",
... | [
"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
} | {
"line": 176,
"column": 14
} | [
{
"pp": "𝕜 : 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 : MeasurableSpace V\nin... | [
"𝕜 : 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 : MeasurableSpace V\ninst✝ : BorelS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 13
} | {
"line": 283,
"column": 14
} | [
{
"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... | [
"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\nG : Type u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 13
} | {
"line": 300,
"column": 14
} | [
{
"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... | [
"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) x‖ ≤ ‖f.toLp 1 volume‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 305,
"column": 8
} | {
"line": 305,
"column": 19
} | {
"line": 305,
"column": 20
} | [
{
"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‖ ≤ ‖... | [
"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) x‖ ≤ ‖f.toLp 1 volume‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 79
} | {
"line": 322,
"column": 4
} | [
{
"pp": "V : 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 (∫ (ξ : ... | [
"V : 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⊢ ∫ (x : V), ↑‖(𝓕 f) x‖ ^ 2 = ∫ (x : V... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.Support | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 13
} | {
"line": 120,
"column": 14
} | [
{
"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",
"ppTerm": "?m.44",
"assigned": true,
"usedConsta... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 55
} | {
"line": 176,
"column": 2
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\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 : ↥(Lp F p μ... | [
"E : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\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 : ↥(Lp F p μ)\ng : 𝓢(E,... | 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": 222,
"column": 4
} | {
"line": 227,
"column": 27
} | {
"line": 229,
"column": 0
} | [
{
"pp": "case h\nE : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\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": 222,
"column": 4
} | {
"line": 227,
"column": 27
} | {
"line": 229,
"column": 0
} | [
{
"pp": "case h\nE : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\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.Analysis.Fourier.Convolution | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 24
} | {
"line": 48,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : NormedAddCommGroup F₂\ninst✝⁷ : NormedAddCommGroup F₃\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\... | [
"𝕜 : Type u_1\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : NormedAddCommGroup F₂\ninst✝⁷ : NormedAddCommGroup F₃\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : Me... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 91
} | {
"line": 222,
"column": 4
} | [
{
"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 = ⊤",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"d : Type u_1\ninst✝¹ : Fintype d\np : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nhp : p ≠ ∞\n⊢ (span ℂ (range (mFourierLp p))).topologicalClosure = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 55
} | {
"line": 288,
"column": 56
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\n⊢ HasSum (fun i ↦ mFourierCoeff (↑↑f) i • mFourierLp 2 i) f",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.to... | [
"d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\n⊢ HasSum (fun i ↦ mFourierCoeff (↑↑f) i • mFourierBasis i) f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 303,
"column": 2
} | {
"line": 304,
"column": 36
} | {
"line": 304,
"column": 37
} | [
{
"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)",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 534,
"column": 2
} | {
"line": 534,
"column": 13
} | {
"line": 534,
"column": 14
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nf : 𝓢(E, F)\ng : 𝓢(E, ℂ)\n⊢ (𝓕 ((toTemperedD... | [
"E : Type u_3\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nf : 𝓢(E, F)\ng : 𝓢(E, ℂ)\n⊢ ∫ (x : E), (𝓕 g) x • f x = ∫... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 54
} | {
"line": 325,
"column": 55
} | [
{
"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‖",
"ppTerm": "?m.83",
"assigned": true,
"use... | [
"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‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 35
} | {
"line": 98,
"column": 8
} | [
{
"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... | [
"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\ninst✝⁵ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 101,
"column": 35
} | {
"line": 101,
"column": 46
} | {
"line": 101,
"column": 47
} | [
{
"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... | [
"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\ninst✝⁵ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 101,
"column": 80
} | {
"line": 101,
"column": 91
} | {
"line": 101,
"column": 92
} | [
{
"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... | [
"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\ninst✝⁵ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Inner | {
"line": 151,
"column": 6
} | {
"line": 151,
"column": 85
} | {
"line": 151,
"column": 86
} | [
{
"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 ↦ ⟪f k₁, f k₂⟫ₙ_[𝕜] = 0\nh✝ : Nonempty ι\n⊢ Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫_[𝕜] = 0",
"ppTerm": "?m.81",
"assigned": false,
"usedConstan... | [
"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 ↦ ⟪f k₁, f k₂⟫ₙ_[𝕜] = 0\nh✝ : Nonempty ι\n⊢ Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫_[𝕜] = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 30
} | {
"line": 66,
"column": 31
} | [
{
"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",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 55
} | {
"line": 86,
"column": 4
} | [
{
"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",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 62
} | {
"line": 186,
"column": 4
} | [
{
"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": 185,
"column": 6
} | {
"line": 185,
"column": 62
} | {
"line": 186,
"column": 4
} | [
{
"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": 185,
"column": 6
} | {
"line": 185,
"column": 62
} | {
"line": 186,
"column": 4
} | [
{
"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.GroupTheory.FiniteAbelian.Basic | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 13
} | {
"line": 245,
"column": 14
} | [
{
"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)... | [
"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) A.toAddSubg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 76
} | {
"line": 68,
"column": 4
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nx y : ℤ\nh : zmod n ↑x = zmod n ↑y\nhn : ↑n ≠ 0\n⊢ ↑x = ↑y",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"ZMod.commRing",
"id",
"Int",
"AddGroupWithOne.toIntCast",
"Nat.cast",
"ZMod",
... | [
"n : ℕ\ninst✝ : NeZero n\nx y : ℤ\nh : zmod n ↑x = zmod n ↑y\nhn : ↑n ≠ 0\n⊢ x ≡ y [ZMOD ↑n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 86,
"column": 50
} | {
"line": 86,
"column": 74
} | {
"line": 86,
"column": 75
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
"ZMod",
"_privat... | [
"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⊢ ∀ (x : ι), f x = g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 13
} | {
"line": 152,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddCommGroup α\na : α\ninst✝ : Finite α\n⊢ (∀ (ψ : AddChar α ℂ), ψ a = 1) ↔ a = 0",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : AddCommGroup α\na : α\ninst✝ : Finite α\n⊢ (∀ (ψ : AddChar α ℂ), ψ a = 1) ↔ a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 13
} | {
"line": 190,
"column": 14
} | [
{
"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",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 13
} | {
"line": 194,
"column": 14
} | [
{
"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",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 114,
"column": 4
} | {
"line": 116,
"column": 12
} | {
"line": 116,
"column": 13
} | [
{
"pp": "case 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 (... | [
"case 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 (x + ↑n) = (∑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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