module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 13
} | {
"line": 124,
"column": 14
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [
"A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : NonnegSpectrum... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 42
} | {
"line": 43,
"column": 43
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nh : MemLp f 2 μ\n⊢ Integrable (fun x ↦ f x ^ 2) μ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"PseudoMetricSpace.toUniformSpace",
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nh : MemLp f 2 μ\n⊢ MemLp (fun x ↦ f x ^ 2) 1 μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 836,
"column": 4
} | {
"line": 836,
"column": 48
} | {
"line": 837,
"column": 6
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn k : ℕ∞\nK : Compacts E\ni : ℕ\nf : 𝓓^{n... | [
"case pos\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn k : ℕ∞\nK : Compacts E\ni : ℕ\nf : 𝓓^{n}_{K}(E, F)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 24
} | {
"line": 139,
"column": 4
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [
"A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : NonnegSpectrum... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 24
} | {
"line": 143,
"column": 4
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [
"A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : NonnegSpectrum... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 154,
"column": 49
} | {
"line": 154,
"column": 60
} | {
"line": 154,
"column": 61
} | [
{
"pp": "A : Type u_2\ninst✝¹⁶ : NonUnitalRing A\ninst✝¹⁵ : StarRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module ℝ A\ninst✝¹² : SMulCommClass ℝ A A\ninst✝¹¹ : IsScalarTower ℝ A A\ninst✝¹⁰ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁹ : PartialOrder A\ninst✝⁸ : StarOrderedRing A\ninst✝⁷... | [
"A : Type u_2\ninst✝¹⁶ : NonUnitalRing A\ninst✝¹⁵ : StarRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module ℝ A\ninst✝¹² : SMulCommClass ℝ A A\ninst✝¹¹ : IsScalarTower ℝ A A\ninst✝¹⁰ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁹ : PartialOrder A\ninst✝⁸ : StarOrderedRing A\ninst✝⁷ : NonnegSpe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 855,
"column": 4
} | {
"line": 855,
"column": 15
} | {
"line": 855,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\nins... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : Normed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 64
} | {
"line": 117,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf g : ↥(Lp E 2 μ)\nx : α\n⊢ ‖⟪↑↑f x, ↑↑g x⟫‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖",
"ppTerm": "?m.65",
"assigned": true,
"usedConstant... | [
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf g : ↥(Lp E 2 μ)\nx : α\n⊢ ‖⟪↑↑f x, ↑↑g x⟫‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖"
] | rw [← @Nat.cast_two ℝ, Real.rpow_natCast, Real.rpow_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 204,
"column": 55
} | {
"line": 204,
"column": 66
} | {
"line": 204,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : NonUnitalRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module 𝕜 A\ninst✝¹² : StarRing A\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : StarOrderedRing A\ninst✝⁹ : IsScalarTower 𝕜 A A\ninst✝⁸ : SMulCommClass 𝕜 A A\ninst✝⁷ : NonUnitalCo... | [
"𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : NonUnitalRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module 𝕜 A\ninst✝¹² : StarRing A\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : StarOrderedRing A\ninst✝⁹ : IsScalarTower 𝕜 A A\ninst✝⁸ : SMulCommClass 𝕜 A A\ninst✝⁷ : NonUnitalContinuousFunc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 46
} | {
"line": 242,
"column": 47
} | [
{
"pp": "A : Type u_2\ninst✝¹⁰ : Ring A\ninst✝⁹ : StarRing A\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : Algebra ℝ A\ninst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : IsTopologicalRing A\ninst✝¹ : T2Space A\ninst✝... | [
"A : Type u_2\ninst✝¹⁰ : Ring A\ninst✝⁹ : StarRing A\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : Algebra ℝ A\ninst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : IsTopologicalRing A\ninst✝¹ : T2Space A\ninst✝ : StarModul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.SmoothApprox | {
"line": 87,
"column": 52
} | {
"line": 87,
"column": 80
} | {
"line": 87,
"column": 81
} | [
{
"pp": "E : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : Chart... | [
"E : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.SmoothApprox | {
"line": 103,
"column": 8
} | {
"line": 103,
"column": 25
} | {
"line": 103,
"column": 26
} | [
{
"pp": "case pos\nE : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst... | [
"case pos\nE : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : Charted... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.SmoothApprox | {
"line": 104,
"column": 8
} | {
"line": 104,
"column": 25
} | {
"line": 104,
"column": 26
} | [
{
"pp": "case neg\nE : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst... | [
"case neg\nE : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : Charted... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 48,
"column": 4
} | {
"line": 48,
"column": 37
} | {
"line": 48,
"column": 38
} | [
{
"pp": "case inl\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\nε : ℝ\nhε : 0 < ... | [
"case inl\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\nε : ℝ\nhε : 0 < ε\nf : E → F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 82
} | {
"line": 102,
"column": 2
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ ... | [
"E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ : Fact (1 ≤ ... | refine (mem_closure_iff_nhds_basis Metric.nhds_basis_closedBall).2 fun ε hε ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 112,
"column": 8
} | {
"line": 113,
"column": 37
} | {
"line": 113,
"column": 38
} | [
{
"pp": "case pos.inl\nα : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.OuterRegular\nhp : p ≠ ∞\ns u : Set α\ns_closed : IsClosed[inst✝⁶... | [
"case pos.inl\nα : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.OuterRegular\nhp : p ≠ ∞\ns u : Set α\ns_closed : IsClosed[inst✝⁶] s\nu_open ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 199,
"column": 38
} | {
"line": 199,
"column": 91
} | {
"line": 199,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : BorelSpace α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure α\ninst✝³ : NormedSpace ℝ E\ninst✝² : R1Space α\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf :... | [
"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : BorelSpace α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure α\ninst✝³ : NormedSpace ℝ E\ninst✝² : R1Space α\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Deriv | {
"line": 65,
"column": 6
} | {
"line": 66,
"column": 79
} | {
"line": 67,
"column": 8
} | [
{
"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\nV : Type u_7\nF : Type u_8\nF₁ : Type u_9\nF₂ : Type u_10\nF₃ : Type u_11\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : RCLike 𝕜\ninst✝ ... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF✝ : Type u_6\nV : Type u_7\nF : Type u_8\nF₁ : Type u_9\nF₂ : Type u_10\nF₃ : Type u_11\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : RCLike 𝕜\ninst✝ : NormedSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Deriv | {
"line": 89,
"column": 6
} | {
"line": 90,
"column": 60
} | {
"line": 90,
"column": 61
} | [
{
"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\nV : Type u_7\nF : Type u_8\nF₁ : Type u_9\nF₂ : Type u_10\nF₃ : Type u_11\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedSpace ℝ E\ninst✝² : RCLike 𝕜\ninst✝¹... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF✝ : Type u_6\nV : Type u_7\nF : Type u_8\nF₁ : Type u_9\nF₂ : Type u_10\nF₃ : Type u_11\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedSpace ℝ E\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 13
} | {
"line": 229,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : BorelSpace α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure α\ninst✝³ : NormedSpace ℝ E\ninst✝² : R1Space α\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\nf : α → E\nε : ℝ\nhε :... | [
"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : BorelSpace α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure α\ninst✝³ : NormedSpace ℝ E\ninst✝² : R1Space α\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\nf : α → E\nε : ℝ\nhε : 0 < ε\nhf :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 41
} | {
"line": 181,
"column": 0
} | [
{
"pp": "T : ℝ\nm n : ℤ\nx : AddCircle T\n⊢ ↑((m + n) • x).toCircle = (fourier m) x * (fourier n) x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"HMul.hMul",
"congrArg",
"Cont... | [] | rw [← fourier_apply]; exact fourier_add | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 41
} | {
"line": 181,
"column": 0
} | [
{
"pp": "T : ℝ\nm n : ℤ\nx : AddCircle T\n⊢ ↑((m + n) • x).toCircle = (fourier m) x * (fourier n) x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"HMul.hMul",
"congrArg",
"Cont... | [] | rw [← fourier_apply]; exact fourier_add | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 31
} | {
"line": 165,
"column": 32
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nG : ι → Type u_4\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝ : DecidableEq ι\ni : ι\na : G i\nf : ↥(lp G 2)\n⊢ ⟪f, lp.single 2 i a⟫ = ⟪↑f i, a⟫",
"ppTerm": "?m.24",
"assigned": false,
"us... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nG : ι → Type u_4\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝ : DecidableEq ι\ni : ι\na : G i\nf : ↥(lp G 2)\n⊢ ⟪f, lp.single 2 i a⟫ = ⟪↑f i, a⟫"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 41
} | {
"line": 270,
"column": 0
} | [
{
"pp": "case e'_9\nT : ℝ\nhT : Fact (0 < T)\np : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nhp : p ≠ ∞\ne_3✝ : Complex.instSemiring = NormedField.toNormedCommRing.toSemiring\ne_6✝ : Lp.instModule ≍ Lp.instModule\n⊢ span ℂ (⇑(toLp p haarAddCircle ℂ) '' range fourier) = span ℂ (⇑↑(toLp p haarAddCircle ℂ) '' range fourier)",
... | [] | simp only [ContinuousLinearMap.coe_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 274,
"column": 6
} | {
"line": 274,
"column": 76
} | {
"line": 274,
"column": 76
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\ni j : ℤ\n⊢ inner ℂ (fourierLp 2 i) (fourierLp 2 j) = if i = j then 1 else 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCom... | [
"T : ℝ\nhT : Fact (0 < T)\ni j : ℤ\n⊢ ∫ (x : AddCircle T), (fourier j) x * (starRingEnd ℂ) ((fourier i) x) ∂haarAddCircle = if i = j then 1 else 0"
] | ContinuousMap.inner_toLp (@haarAddCircle T hT) (fourier i) (fourier j) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 290,
"column": 38
} | {
"line": 290,
"column": 91
} | {
"line": 290,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.WeaklyRegular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : MemLp f (ENNReal.ofReal p) μ\nε : ℝ\nhε... | [
"α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.WeaklyRegular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : MemLp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 215,
"column": 2
} | {
"line": 221,
"column": 12
} | {
"line": 223,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : Orthog... | [] | rw [hV.linearIsometry_apply, ← tsum_ite_eq i (fun _ ↦ V i x)]
congr
ext j
rw [lp.single_apply]
split_ifs with h
· subst h; simp
· simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 215,
"column": 2
} | {
"line": 221,
"column": 12
} | {
"line": 223,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : Orthog... | [] | rw [hV.linearIsometry_apply, ← tsum_ite_eq i (fun _ ↦ V i x)]
congr
ext j
rw [lp.single_apply]
split_ifs with h
· subst h; simp
· simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 242,
"column": 4
} | {
"line": 247,
"column": 80
} | {
"line": 249,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[𝕜]... | [] | apply topologicalClosure_minimal
· refine iSup_le ?_
rintro i x ⟨x, rfl⟩
use lp.single 2 i x
exact hV.linearIsometry_apply_single x
exact hV.linearIsometry.isometry.isUniformInducing.isComplete_range.isClosed | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 13
} | {
"line": 314,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.WeaklyRegular\nf : α → E\nε : ℝ\nhε : 0 < ε\nhf : MemLp f (ENNReal.ofReal 1) μ\n⊢ ∃ g, ∫ ... | [
"α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.WeaklyRegular\nf : α → E\nε : ℝ\nhε : 0 < ε\nhf : MemLp f (ENNReal.ofReal 1) μ\n⊢ ∃ g, ∫ (x : α), ‖f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 242,
"column": 4
} | {
"line": 247,
"column": 80
} | {
"line": 249,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[𝕜]... | [] | apply topologicalClosure_minimal
· refine iSup_le ?_
rintro i x ⟨x, rfl⟩
use lp.single 2 i x
exact hV.linearIsometry_apply_single x
exact hV.linearIsometry.isometry.isUniformInducing.isComplete_range.isClosed | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 44
} | {
"line": 323,
"column": 45
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : AddCircle T → E\nhf : Integrable f haarAddCircle\nhg : Integrable g haarAddCircle\nx : ℤ\n⊢ fourierCoeff (f + g) x = (fourierCoeff f + fourierCoeff g) x",
"ppTerm": "?m.55",
"assigned": true,
... | [
"T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : AddCircle T → E\nhf : Integrable f haarAddCircle\nhg : Integrable g haarAddCircle\nx : ℤ\n⊢ ∫ (t : AddCircle T), (fourier (-x)) t • f t + (fourier (-x)) t • g t ∂haarAddCircle =\n ∫ (t : AddCircle T), (fourier ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 332,
"column": 30
} | {
"line": 332,
"column": 41
} | {
"line": 332,
"column": 42
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nι : Type u_2\nf : ι → AddCircle T → E\na : ι\ns : Finset ι\nha : a ∉ s\niha : (∀ i ∈ s, Integrable (f i) haarAddCircle) → fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)\nhf : ∀ i ∈ insert a s, Int... | [
"T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nι : Type u_2\nf : ι → AddCircle T → E\na : ι\ns : Finset ι\nha : a ∉ s\niha : (∀ i ∈ s, Integrable (f i) haarAddCircle) → fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)\nhf : ∀ i ∈ insert a s, Integrable (f i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 287,
"column": 30
} | {
"line": 287,
"column": 80
} | {
"line": 287,
"column": 81
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nF : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace ↥(F i)\nhFortho : OrthogonalFamily 𝕜 (fun i ↦ ↥(F i)) fun i ↦ (F i).subtypeₗᵢ\nhFtotal : ⊤ ≤... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nF : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace ↥(F i)\nhFortho : OrthogonalFamily 𝕜 (fun i ↦ ↥(F i)) fun i ↦ (F i).subtypeₗᵢ\nhFtotal : ⊤ ≤ (⨆ i, F i).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 327,
"column": 65
} | {
"line": 334,
"column": 68
} | {
"line": 337,
"column": 0
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nι : Type u_2\ns : Finset ι\nf : ι → AddCircle T → E\nhf : ∀ i ∈ s, Integrable (f i) haarAddCircle\n⊢ fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)",
"ppTerm": "?m.42",
"assigned": true,
... | [] | by
classical
induction s using Finset.induction_on with
| empty => ext; simp [fourierCoeff]
| insert a s ha iha =>
obtain ⟨hf₁, hf₂⟩ := by simpa using hf
rw [s.sum_insert ha, s.sum_insert ha,
fourierCoeff.add hf₁ (integrable_finsetSum' s hf₂), iha hf₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 441,
"column": 47
} | {
"line": 441,
"column": 58
} | {
"line": 441,
"column": 59
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nb : HilbertBasis ι 𝕜 E\nx : E\n⊢ HasSum (fun i ↦ ↑(b.repr x) i • b i) x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"LinearIsometryEquiv.instEqu... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nb : HilbertBasis ι 𝕜 E\nx : E\n⊢ HasSum (fun i ↦ ↑(b.repr x) i • b i) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 480,
"column": 6
} | {
"line": 480,
"column": 99
} | {
"line": 481,
"column": 8
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝ : Fintype ι\nb : HilbertBasis ι 𝕜 E\nthis : IsClosed[Pseudo... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝ : Fintype ι\nb : HilbertBasis ι 𝕜 E\nthis : IsClosed[PseudoMetricSpace.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 491,
"column": 2
} | {
"line": 491,
"column": 86
} | {
"line": 492,
"column": 4
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\nb : HilbertBasis ι 𝕜 ↥U\nx : E\n⊢ HasSum (fun i ↦ ⟪↑(b i), x⟫ • b i) (U.orthogonalProjectionOnto x)",
"ppTerm": "?m.34",
... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\nb : HilbertBasis ι 𝕜 ↥U\nx : E\n⊢ HasSum (fun i ↦ ⟪↑(b i), x⟫ • b i) (U.orthogonalProjectionOnto x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 559,
"column": 8
} | {
"line": 560,
"column": 73
} | {
"line": 560,
"column": 74
} | [
{
"pp": "𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nw : Set E\nhws : w ⊇ s\nhw_ortho : Orthonormal 𝕜 Subtype.val\nhw_max : ∀ u ⊇ w, Orthonormal 𝕜 Subtype.val → u = w\n⊢ (s... | [
"𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nw : Set E\nhws : w ⊇ s\nhw_ortho : Orthonormal 𝕜 Subtype.val\nhw_max : ∀ u ⊇ w, Orthonormal 𝕜 Subtype.val → u = w\n⊢ (span 𝕜 w)ᗮ =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 13
} | {
"line": 545,
"column": 14
} | [
{
"pp": "T : ℝ\nn : ℤ\nx : ℝ\n⊢ HasDerivAt (fun y ↦ (fourier (-n)) ↑y) (-2 * ↑π * I * ↑n / ↑T * (fourier (-n)) ↑x) x",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Int.cast",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"N... | [
"T : ℝ\nn : ℤ\nx : ℝ\n⊢ HasDerivAt (fun y ↦ (starRingEnd ℂ) (cexp (2 * ↑π * I * ↑n * ↑y / ↑T)))\n (-(2 * ↑π * I * ↑n) / ↑T * (starRingEnd ℂ) (cexp (2 * ↑π * I * ↑n * ↑x / ↑T))) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 553,
"column": 16
} | {
"line": 553,
"column": 34
} | {
"line": 553,
"column": 34
} | [
{
"pp": "case e'_8\nT : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn : n ≠ 0\nx y : ℝ\n⊢ ↑T / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑y = (fourier (-n)) ↑y / (-2 * ↑π * I * ↑n / ↑T)",
"ppTerm": "?e'_8",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing... | [
"case e'_8\nT : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn : n ≠ 0\nx y : ℝ\n⊢ ↑T / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑y = (fourier (-n)) ↑y * ↑T / (-2 * ↑π * I * ↑n)"
] | div_div_eq_mul_div | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 576,
"column": 4
} | {
"line": 576,
"column": 15
} | {
"line": 576,
"column": 16
} | [
{
"pp": "a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\nthis : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)\n⊢ ↑b = ↑a... | [
"a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\nthis : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)\n⊢ ↑b = ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 52
} | {
"line": 78,
"column": 53
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\na : E\nhs : MeasurableSet s\nh'st : t ∈ 𝓝[s] x₀\nhlφ... | [
"α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\na : E\nhs : MeasurableSet s\nh'st : t ∈ 𝓝[s] x₀\nhlφ : ∀ (u : Se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransform | {
"line": 192,
"column": 52
} | {
"line": 210,
"column": 8
} | {
"line": 212,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝²² : CommRing 𝕜\nV : Type u_2\ninst✝²¹ : AddCommGroup V\ninst✝²⁰ : Module 𝕜 V\ninst✝¹⁹ : MeasurableSpace V\nW : Type u_3\ninst✝¹⁸ : AddCommGroup W\ninst✝¹⁷ : Module 𝕜 W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace ℂ E\ninst✝¹⁴ :... | [] | by
rw [integral_integral_swap]
have : Integrable (fun (p : W × V) ↦ ‖M‖ * (‖g p.1‖ * ‖f p.2‖)) (ν.prod μ) :=
(hg.norm.mul_prod hf.norm).const_mul _
apply this.mono
· change AEStronglyMeasurable (fun p : W × V ↦ (M (g p.1) (e (-(L p.2) p.1) • f p.2))) _
have A : AEStronglyMeasurable (fun (p : W × V) ↦ e ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 164,
"column": 8
} | {
"line": 164,
"column": 56
} | {
"line": 164,
"column": 57
} | [
{
"pp": "case hbc\nα : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\nhs : MeasurableSet s\nht : MeasurableSet t\... | [
"case hbc\nα : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\nhs : MeasurableSet s\nht : MeasurableSet t\nhts : t ⊆ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 67
} | {
"line": 104,
"column": 68
} | [
{
"pp": "x : ℝ\nh1 : ∀ (y : ℝ), ↑(𝐞 y) = (fourier 1) ↑y\n⊢ HasDerivAt (fun x ↦ ↑(𝐞 x)) (2 * ↑π * I * ↑(𝐞 x)) x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSemin... | [
"x : ℝ\nh1 : ∀ (y : ℝ), ↑(𝐞 y) = (fourier 1) ↑y\n⊢ HasDerivAt (fun x ↦ (fourier 1) ↑x) (2 * ↑π * I * (fourier 1) ↑x) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 234,
"column": 33
} | {
"line": 234,
"column": 63
} | {
"line": 234,
"column": 64
} | [
{
"pp": "E : 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 : BorelSpace V\ninst✝... | [
"E : 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 : BorelSpace V\ninst✝ : SecondCou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 233,
"column": 8
} | {
"line": 233,
"column": 43
} | {
"line": 233,
"column": 44
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\nφ : ι → α → ℝ\na : E\ninst✝ : CompleteSpace E\nt : Set α\nht : MeasurableSet t\n... | [
"α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\nφ : ι → α → ℝ\na : E\ninst✝ : CompleteSpace E\nt : Set α\nht : MeasurableSet t\nh'ts : t ∈ �... | 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.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": 59,
"column": 2
} | {
"line": 59,
"column": 42
} | {
"line": 59,
"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": 62,
"column": 2
} | {
"line": 62,
"column": 39
} | {
"line": 62,
"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": 75,
"column": 2
} | {
"line": 75,
"column": 13
} | {
"line": 75,
"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": 80,
"column": 2
} | {
"line": 80,
"column": 93
} | {
"line": 81,
"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": 94,
"column": 4
} | {
"line": 94,
"column": 15
} | {
"line": 94,
"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": 95,
"column": 2
} | {
"line": 95,
"column": 49
} | {
"line": 95,
"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": 99,
"column": 2
} | {
"line": 101,
"column": 9
} | {
"line": 101,
"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": 106,
"column": 52
} | {
"line": 106,
"column": 63
} | {
"line": 106,
"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": 108,
"column": 45
} | {
"line": 108,
"column": 56
} | {
"line": 108,
"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": 112,
"column": 2
} | {
"line": 113,
"column": 9
} | {
"line": 113,
"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": 133,
"column": 2
} | {
"line": 133,
"column": 13
} | {
"line": 133,
"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": 181,
"column": 4
} | {
"line": 181,
"column": 30
} | {
"line": 181,
"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": 175,
"column": 2
} | {
"line": 183,
"column": 41
} | {
"line": 185,
"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": 175,
"column": 2
} | {
"line": 183,
"column": 41
} | {
"line": 185,
"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.PeakFunction | {
"line": 299,
"column": 4
} | {
"line": 318,
"column": 21
} | {
"line": 319,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCo... | [
"α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCompact s\nhμ ... | have M : ∀ n, ∀ x ∈ s \ u, φ n x ≤ (μ.real (v ∩ s))⁻¹ * (t / t') ^ n := by
intro n x hx
have B : t' ^ n * μ.real (v ∩ s) ≤ ∫ y in s, c y ^ n ∂μ :=
calc
t' ^ n * μ.real (v ∩ s) = ∫ _ in v ∩ s, t' ^ n ∂μ := by simp [mul_comm]
_ ≤ ∫ y in v ∩ s, c y ^ n ∂μ := by
apply set... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 38
} | {
"line": 166,
"column": 39
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : 𝓢(E, F)\ninst✝ : ProperSpace E\nk : ℤ\n⊢ ⇑f =O[cocompact E] fun x ↦ ‖x‖ ^ k",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"us... | [
"E : Type u_5\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : 𝓢(E, F)\ninst✝ : ProperSpace E\nk : ℤ\n⊢ ⇑f =O[cocompact E] fun x ↦ ‖x‖ ^ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 549,
"column": 18
} | {
"line": 549,
"column": 29
} | {
"line": 549,
"column": 30
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : 𝓢(E, F)\nn : ℕ\nC : ℝ\nCpos : 0 < C\nhC : ∀ (x : E), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C\n⊢ ∀ (x : E), ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C * (1 + ‖x‖) ^ 0"... | [
"E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : 𝓢(E, F)\nn : ℕ\nC : ℝ\nCpos : 0 < C\nhC : ∀ (x : E), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C\n⊢ ∀ (x : E), ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | 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.Distribution.SchwartzSpace.Basic | {
"line": 655,
"column": 93
} | {
"line": 657,
"column": 78
} | {
"line": 658,
"column": 4
} | [
{
"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✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedField 𝕜\ninst✝³ : NormedAddCommGroup G\ninst... | [] | by
gcongr
exact norm_iteratedFDeriv_clm_apply_const (f.smooth _).contDiffAt le_rfl | [anonymous] | Lean.Parser.Term.byTactic |
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.MeasureTheory.Integral.PeakFunction | {
"line": 365,
"column": 32
} | {
"line": 365,
"column": 53
} | {
"line": 367,
"column": 0
} | [
{
"pp": "case h₂\nα : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝³ : CompleteSpace E\ninst✝² : MetrizableSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\... | [] | apply interior_subset | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
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.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.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.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.MeasureTheory.Integral.PeakFunction | {
"line": 414,
"column": 6
} | {
"line": 414,
"column": 17
} | {
"line": 414,
"column": 18
} | [
{
"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": 821,
"column": 4
} | {
"line": 821,
"column": 38
} | {
"line": 822,
"column": 6
} | [
{
"pp": "case pos\n𝕜 : Type u_2\nE : 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 : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAlgebra ℝ 𝕜\ninst✝ : NormedSpace 𝕜 F\ng : E → 𝕜\nf : 𝓢(E, F)\nhg : Function.HasT... | [
"case pos\n𝕜 : Type u_2\nE : 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 : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAlgebra ℝ 𝕜\ninst✝ : NormedSpace 𝕜 F\ng : E → 𝕜\nf : 𝓢(E, F)\nhg : Function.HasTemperateGrow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 313,
"column": 28
} | {
"line": 313,
"column": 66
} | {
"line": 313,
"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.Distribution.SchwartzSpace.Basic | {
"line": 862,
"column": 28
} | {
"line": 862,
"column": 39
} | {
"line": 862,
"column": 40
} | [
{
"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✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAlgebra... | [
"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✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAlgebra ℝ 𝕜\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 386,
"column": 19
} | {
"line": 386,
"column": 29
} | {
"line": 386,
"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.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.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": 898,
"column": 17
} | {
"line": 901,
"column": 14
} | {
"line": 903,
"column": 0
} | [
{
"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✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAlgebra... | [] | by
simp only [Finset.sup_insert, schwartzSeminormFamily_apply, Finset.sup_singleton,
Seminorm.coe_sup, Pi.sup_apply]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 84
} | {
"line": 434,
"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": 439,
"column": 2
} | {
"line": 440,
"column": 33
} | {
"line": 440,
"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.Gamma.Basic | {
"line": 519,
"column": 6
} | {
"line": 519,
"column": 17
} | {
"line": 519,
"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.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.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 90
} | {
"line": 67,
"column": 4
} | [
{
"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",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"C... | [
"case inl\np s : ℝ\nhs : -1 < s\nhp✝ : 1 ≤ p\nhp : 1 < p\nh_exp : ∀ (x : ℝ), ContinuousAt (fun x ↦ rexp (-x)) x\n⊢ IntegrableOn (fun x ↦ x ^ s * rexp (-x ^ p)) (Ioi 0) volume"
] | 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.Fourier.FourierTransformDeriv | {
"line": 846,
"column": 40
} | {
"line": 846,
"column": 51
} | {
"line": 846,
"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.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.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 35
} | {
"line": 131,
"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": 135,
"column": 2
} | {
"line": 135,
"column": 13
} | {
"line": 135,
"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": 146,
"column": 2
} | {
"line": 146,
"column": 13
} | {
"line": 146,
"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": 154,
"column": 2
} | {
"line": 154,
"column": 13
} | {
"line": 154,
"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.Gaussian.GaussianIntegral | {
"line": 219,
"column": 18
} | {
"line": 219,
"column": 79
} | {
"line": 220,
"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": 219,
"column": 18
} | {
"line": 219,
"column": 79
} | {
"line": 220,
"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": 219,
"column": 18
} | {
"line": 219,
"column": 79
} | {
"line": 220,
"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": 233,
"column": 6
} | {
"line": 233,
"column": 62
} | {
"line": 233,
"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 |
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